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You correctly solved the first part of the problem. Your answer tells you that this equation:
.
y - 5 = 0(x - (-4))
.
will always be true if y = +5 because when you substitute +5 for y this equation becomes:
.
0 = 0(x - (-4))
.
which reduces to:
.
0 = 0
.
Notice that it does not matter what value you assign to x. x could be -100, +50, or any other value you choose. As long as y is +5, the slope of the graph of this equation will be zero.
.
How does this translate to a graph? Any time that you are told that a line has a slope of zero, you can immediately say that it's graph is a horizontal line because its value of y never changes as you move to the right along the x-axis.
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In this problem, you calculated that the unchanging value of y is +5. Therefore, you can describe the graph as being horizontal (parallel to the x-axis) and crossing the y-axis at +5. In picture form:
.

.
The horizontal red line above the x-axis is the graph of a line having a zero slope and a y value of +5. Note that on the x-axis at the point where x equals -4, the corresponding value of y as shown by the graph is +5, so the two original constraints (a zero slope graph having the x,y point (-4,5) on the line) are both satisfied.
.
Hope this helps.

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This is a very common size for a right triangle. If a right triangle has a hypotenuse of 5 and one of its legs is 4, then the other leg is 3. Other than knowing this, you can find the unknown leg by using the Pythagorean theorem. It says that the square of the hypotenuse (in this case 5) equals the sum of the squares of the two legs. One of the legs in this case is known to be 4 and the other leg is unknown. Call this unknown leg A. So we can write:
.

.
Square the two numbers and this equation becomes:
.

.
Subtract 16 from both sides of the equation to reduce it to:
.

.
Solve for A by taking the square root of both sides and you have:
.

.
So now you know all three sides are 3, 4, and 5 in which the hypotenuse (or longest side) is 5.
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This means that the base (given in the problem) is 4 and the perpendicular to it (the altitude in a right triangle) is the leg that you found to be 3.
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The area of a triangle is 1/2 times the base times the altitude. So the area of this triangle is:
.

.
Find the area by multiplying the three numbers on the right side to get:
.

.
And don't forget to add the units of square inches since the problem said that the two given sides in the triangle had units of inches.
.
Hope this helps you to understand the problem.
.

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The equation that applies to this problem is:
.
Distance = Speed * Time
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The first thing to do is to ensure that the units are consistent. In this problem the distance you are given is in "miles" and the speed also involves "miles". This means that you do not have to make any changes or conversionss to get the units to agree. Then since the speed involves "hours", the answer for the time will also have the unit "hours".
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Just substitute 18 for the distance in the equation and 72 for the speed to get:
.
18 = 72*Time
.
Solve for Time by dividing both sides of this equation by 72:
.
18/72 = (72/72)* Time
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The 72/72 is just 1. This reduces the equation to:
.
18/72 = Time
.
The fraction on the left side of this equation can be reduced by either dividing 72 into 18 to get 0.25 (which is equal to 1/4) or by replacing 72 by its equivalent value of 18*4. In the latter case this makes the equation become:
.
18/(18*4) = Time
.
Then cancel the 18 in the numerator with the 18 in the denominator and this reduces the equation to:
.
1/4 = Time
.
And the units is hours. So the answer is:
.
Time = 1/4 hours
.
or in an equivalent form:
.
Time = 0.25 hours
.
At a speed of 72 mph, you go 18 miles in a-quarter of an hour (which is 15 minutes).
.
Hope this helps you to understand the problem a little better.
.

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The expression you are given is 5xy, and this means 5 times x times y.
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To evaluate the expression for x = 2 and y = 3, just substitute 2 for x and 3 for y in the expression. When you make those substitutions you get:
.
5 times 2 times 3
.
5 times 2 is 10 and then you multiply that by 3 to get the answer 30.
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So you can say that when x 2 and y = 3, then 5xy = 30
.
Hope this helps you to understand this problem.

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Given the following three equations to solve simultaneously:
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(A) 3x + 4y + z = 14
(B) .. + 2y + 7z = 24
(C) .. - 2y + 3z = 5
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Begin by adding equations B and C vertically. Note that the + 2y and the - 2y cancel each other out. The + 7z and the + 3z add to give + 10z and on the other side of the equal sign the 24 and 5 add up to 29. So the result of adding these two equations is the new equation:
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10z = 29
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Solve this new equation for z by dividing both sides by 10 to get:
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z = 29/10 = 2.9
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Since you now know that z = 2.9, you can return to either equation B or equation C and, after substituting 2.9 for z, you can solve for y. Go to equation B and substitute 2.9 for z to get:
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2y + 7*2.9 = 24
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Multiply out the 7 times 2.9 to get 20.3 and the equation becomes:
.
2y + 20.3 = 24
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Subtract 20.3 from both sides of this equation and you are left with:
.
2y = 3.7
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Solve for y by dividing both sides of this equation by 2 to get:
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y = 3.7/2 = 1.85
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Now you know that z = 2.9 and y = 1.85. Equation A is the only equation that has a term containing x. So you next go to equation A and substitute 2.9 for z and 1.85 for y to get:
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3x + 4(1.85) + 2.9 = 14
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Multiply the 4 times 1.85 and the equation becomes:
.
3x + 7.4 + 2.9 = 14
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Add the two numbers on the left side:
.
3x + 10.3 = 14
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Subtract 10.3 from both sides:
.
3x = 3.7
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Solve for x by dividing both sides by 3 to get:
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x = 3.7/3 = 1.23333333...
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So the values of x, y, and z that satisfy all three of the given equations simultaneously are x = 1.23333333..., y = 1.85, and z = 2.9
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Hope this helps you to understand the problem a little better. Check my work to ensure that it has no errors. You can also practice a little by using 2.9 for z and substituting that value into equation C. Then solve for y and see if you again get 1.85 for y just as you did by substituting 2.9 into equation B above.

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Given to divide:
.

.
Note that this is one great big fraction. The numerator is:
.

.
and the denominator is:
.

.
So you are dividing the numerator by the denominator. Remember the rule for dividing by a fraction ... namely that you can divide by a fraction by inverting the fractional divisor and then multiplying that inverted fraction by the number you are dividing into.
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When you invert the denominator you get:
.

.
Now multiply that inverted fraction times the numerator and you have:
.

.
Note that the denominator of the first fraction is and it will divide into the numerator of the second fraction to give just 2. This reduces the problem to:
.

.
Next note that the denominator of the second fraction can be factored by 9 to give:
.

.
Then the numerator of the first fraction can be factored by 3 to result in:
.

.
So you have a (y+4) factor in the numerator and a (y+4) factor in the denominator. These cancel out and you are left with:
.

.
which after multiplication gives:
.

.
and 6 divided by 9 is equal to
.
and that is the answer to this problem.
.
I hope this helps you to see your way through this problem
.

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Given to solve by factoring:
.

.
First, square the terms in the parentheses by multiplying:
.

.
When you do this multiplication you get:
.

.
Substitute this into the original problem in place of and the problem then becomes:
.

.
Combine the two x-squared terms to get:
.

.
Subtract 650 from both sides and you have:
.

.
You can simplify this a little by dividing both sides (all terms) by 2 to get:
.

.
Now you can do the factoring process. The negative sign on the 323 tells you that you will have one positive factor and one negative factor whose product gives you negative 323. The coefficient of the x term is +2. This tells you that the positive factor must be bigger than the negative factor by 2.
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Now you can make some educated guesses. If the two factors were +20 and -18, their product would be (you can do this in your head) -360. You need two numbers whose product is -323, so you are pretty close, but the factors you guessed are a little too big because they give a product of -360. How about trying +18 and -16. The product of these two numbers is -288 and since this is smaller that -323, you need two bigger numbers. If you now try +19 and -17, you will find that their product is -323, just as you need it to be. So you know that the product of the two factors is:
.

.
and if you set these equal to zero as above you have:
.

.
Notice that this equation will be true if either of the factors equals zero. So you can set both factors equal to zero and solve for x to get:
.

.
and solve for x to get x = -19
.
Then:
.

.
and solve for x to get x = +17
.
Those are the two solutions for x in the equation:
.

.
You can check the solutions by first substituting -19 for x and you have:
.

.
When you square -19 you get +361. and when you add -19 and +2 in the parentheses you get -17. Then square that and you have get +289. Substituting these values into the equation gives:
.

.
And if you add the terms on the left side you find that the equation balances, so x = -19 works.
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You can do the same kind of analysis for the second answer. Go back to the original equation and substitute +17 for x and you find that it also checks out.
.
Hope this helps you see where you went slightly off track and lets you see how the answers can be found.
.

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Let J equal the number of points that James gets, A equal the number of points that Ashley gets, and N equal the number of points that Nicole gets.
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The sum of all the points is 340. Therefore, we can write the equation:
.
J + A + N = 340
.
But we are told that the number of points that James gets is 40 less than the number of points Ashley gets. So if we take 40 points from Ashley the result equals the number of points for James. In equation form this says:
.
J = A - 40
.
This tells us that in our equation for the total number of points we can replace J with A - 40 because they are equal. So we substitute A - 40 for J and the equation for the total number of points becomes:
.
(A - 40) + A + N = 340
.
Next we are told that the number of points that Ashley gets is 80 less than the number of points Nicole gets. So if we take 80 points from Nicole, the result is the number of points for Ashley. In equation form this is:
.
A = N - 80
.
This means that we can replace every A in our equation with N - 80 because they are equal. So let's do that and our equation then is:
.
((N - 80) - 40) + (N - 80) + N = 340
.
Since all the sets of parentheses are preceded by plus signs, we can just remove the parentheses without changing the signs of the terms they contain. Removing the parentheses results in:
.
N - 80 - 40 + N - 80 + N = 340
.
The constants on the left side are -80, -40, and -80. When we add these we get -200 and our equation becomes:
.
N + N + N - 200 = 340
,
Combine the three N terms to make our equation:
.
3N - 200 = 340
.
Get rid of the -200 on the left side by adding 200 to both sides. This results in:
.
3N = 540
.
Finally, solve for N, the number of points Nicole gets, by dividing both sides by 3 and we get:
.
N = 180
.
Ashley gets 80 points less than Nicole, so Ashley gets 100 points, and James gets 40 points less than Ashley, so he gets 60 points.
.
That's the solution. James = 60, Ashley = 100, and Nicole = 180 points.
.
To check this answer, we just add the three scores (60, 100, and 180) and ensure that the total is 340 points. It is, so our answer is correct.
.
Hope this helps you to understand the problem.
.

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Given to solve:
.

.
Square both sides to get:
.

.
add 9 to both sides to get rid of the -9 on the left side:
.

.
Take the square root of both sides to get the two answers:
.
and
.
You can check these two answers by returning to the original equation of:
.

.
and first substituting +5 for x to get:
.

.
Square the 5 and this equation becomes:
.

.
Do the subtraction on the left side to get:
.

.
4 is the square root of 16, so this equation reduces to:
.

.
And since this is true, we know that if x = +5, the equation works and therefore +5 is a solution.
.
Next, return to the original equation of:
.

.
and this time substitute -5 for x to get:
.

.
Square the -5 to again get +25 and this equation becomes:
.

.
Just like last time do the subtraction on the left side to get:
.

.
Again, 4 is the square root of 16, so this equation reduces to:
.

.
And since this is true, we know that if x = -5, the equation works and therefore -5 is also a solution.
.
Hope this helps you to understand the problem a little more and you can see how to solve and check a problem such as this one.
.

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The general idea of solving linear equations such as this one is to do what is necessary to get the unknown variable on the left side of the equal sign and the numbers on the right side. Looking at the problem you are given should tell you that you have to do a little work to do that.
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First, note that on the left side you have a distributed multiplication, so you need to multiply that out. Do this multiplication by multiplying the 4 times each of the terms in the parentheses. 4 times m is 4m and 4 times -3 is -12. So after the multiplication, the equation becomes:
.
4m - 12 = m + 6
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Now we are ready to collect all the terms containing m on the left side of the equal sign and all the numbers on the right side. Remember that whatever we do to one side of an equation we must also do to the other side to keep the equality unchanged.
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Let's get rid of the m on the right side by subtracting m on the right side. But if we do that we must also subtract an m from the left side. This subtraction results in:
.
4m - m - 12 = m - m + 6
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On the left side the 4m - m results in 3m and on the right side the m - m results in zero, meaning that the m is gone on the right side. So our equation is then:
.
3m - 12 = 6
.
Now let's get rid of the -12 on the left side. We can do that by adding +12 to the left side, but then we must also add +12 to the right side:
.
3m - 12 + 12 = 6 + 12
.
On the left side the -12 and the + 12 add to zero, so the -12 has disappeared from the left side. On the right side the 6 + 12 adds to 18. What's left is:
.
3m = 18
.
Finally, our goal is to solve for m. On the left side we have 3m and we can make that just m if we divide the left side by 3. But to do that we must also divide the right side by 3. This is shown as follows:
.
3m/3 = 18/3
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And when we do these divisions we get the answer:
.
m = 6
.
That's the answer to the problem. We can check it by returning to the original problem and substituting 6 for m as follows:
.
4(m - 3) = m + 6
.
Now let m be 6:
.
4(6 - 3) = 6 + 6
.
Do the subtraction in the parentheses of 6 - 3 which gives 3:
.
4(3) = 6 + 6
.
Multiply out the left side (4 times 3 is 12) and do the addition on the right side ( 6 + 6 = 12). This results in:
.
12 = 12
.
And since the left side and right side are both 12 when m is 6, we know we have solved the problem correctly.
.
When m equals 6 the equation is equal on both sides.
.
Hope this helps you to have a little better understanding of what can be done to solve equations such as this.
.

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Let's define:
.
For part (a) the annual linear total cost function as C(x)
For part (b) the annual linear total revenue function as R(x) and
For part (c) the annual linear profit function as P(x)
.
In part a, the cost of production will increase by $23 for each unit that is produced. So to find the cost for the units produced, you just multiply the cost per unit times the number of units that are made. That is $23 times x or just 23x. But the total cost also includes the annual fixed costs of $400,000 probably consisting of taxes on the property, average annual building maintenance costs, average annual utility costs, and other similar costs that are incurred whether or not you produce any products. The total annual costs are the sum of these two. So we write the annual cost function in dollars as follows:
.
C(x) = 23x + 400,000
.
That's the answer to part (a)
.
For part (b) the total annual revenue function is the entire amount of money that the company receives for selling units of the product during the year. Since the selling price is is $55 per unit, and the company sells x units during the year, the total revenue for the year is $55 times x. We can express this as follows:
.
R(x) = 55x
.
and that's the answer to part (b)
.
But all the revenue is not profit. The profit is found by subtracting from the income (total revenue) the cost of production. In other words the annual profits is calculated from:
.
P(x) = R(x) - C(x)
.
But we now know what R(x) and C(x) are and we can substitute those relationships into this equation as follows:
.
P(x) = 55x - (23x + 400,000)
.
Since the parentheses are preceded by a minus sign, we can remove them if we change the signs of each of the terms within the parentheses. This makes the equation become:
.
P(x) = 55x - 23x - 400,000
.
We can then combine the 55x and -23x to get 32x and the equation is then:
.
P(x) = 32x - 400,000
.
And that's the answer to part (c)
.
Just as an interesting exercise, we weren't asked but we might want to find how many units the company has to sell each year, just to break even. By breaking even we mean that the company neither loses nor makes any money for the year. At this point the profit would be zero. So we can go to the profit equation and set P(x) equal to zero. The equation then becomes:
.
0 = 32x - 400,000
.
Subtract 32x from both sides and you have:
.
-32x = -400,000
.
Solve for x by dividing both sides by -32 and you get:
.
x = -400,000/-32 = 12,500
.
This means that unless the company sells 12,500 units in a year, it will lose money. Selling more than 12,500 units will make the company profitable.
Hope this helps you to understand the problem and how it can be worked.

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The fact that the exponent on the 10 is NEGATIVE tells you to move the decimal point to the LEFT. The fact that the exponent is 2 tells you to move the decimal point two places. So combined, this means that the decimal point gets moved two places to the left.
.
So start with 5.93
.
After you move the decimal point to the left one place you would have 0.593. Then you have to move the decimal point to the left again so that you have moved it a total of 2 places to the left. When you do that you have 0.0593.
.
And that's the answer ... answer (a) in your answer list.
.
In summary, when the exponent on the 10 is negative, move the decimal point that many places to the left. If the exponent on the 10 is positive, move the decimal point that many places to the right.
.
Hope this helps to clear things up for you.
.

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Let x represent the unknown amount invested in the home owners' cooperative.
.
Since 50 000 more is invested in the teachers' cooperative, we can write that the investment in the teachers' cooperative is x + 50 000
,
The sum of these two investments is:
.
x + x + 50 000 and adding the two x terms results in a total of 2x + 50 000
.
But the problem tells you that the sum of the two investments is 110 000. Therefore, we can write the equation:
.
2x + 50 000 = 110 000
.
To solve this for x we can first eliminate the 50 000 on the left side by subtracting 50 000 from both sides to get:
.
2x = 60 000
.
Now we can solve for x by dividing both sides of this equation by 2 to get:
.
x = 30 000
.
Since we chose x to represent the amount invested in the home owners' cooperative, we know that we have 30 000 in it. And since 50 000 more was invested in the teachers' cooperative we know that we have invested 30 000 plus 50 000 for a total of 80 000 in the teachers cooperative.
.
And as a check, the total of 30 000 in the home owners' cooperative plus the 80 000 in the teachers' cooperative does add up to the total 110 000 that we have invested all together.
.
I hope this helps you to understand the problem and how you can solve it.
.

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The inequalities involving triangles tell you that the sum of the two shortest sides must be longer than the third side or else a triangle cannot be formed.
.
Since you are not given the length of T, the third side, it can vary in length.
.
Since T can vary in length, if it is drawn so that it is the longest side, then the two short sides are 5x - 2 and 6x + 7. The sum of these two short sides is:
.
(5x - 2) + (6x + 7)
.
And when you add them together you get 11x + 5. This sum must be greater than T, so we can write:
.
11x + 5 > T
.
which is read either as 11x + 5 must be greater than T or T must be less than 11x + 5.
.
Let's write it as T must be less than 11x + 5:
.
T < 11x + 5
.
At this point, notice that only answers (a) and (b) involve 11x + 5 being greater than T. So we can eliminate choices (c) and (d).
.
Now let's imagine that we make the length of T smaller and smaller until it is the smallest side in the triangle. This would make the two small sides be T and 5x -2. In order for a triangle to be formed, the sum of these two small sides must be greater than the third side which is now 6x + 7. Let's first add the two small sides:
.
T + (5x - 2) which can be written as just T + 5x - 2
.
and since this sum must be greater than the third side which is 6x + 7 we can write this inequality as:
.
T + 5x - 2 > 6x + 7
.
We want to solve this inequality for T, so we need to get rid of the 5x and the minus 2 on the left side of the inequality sign. We can do that by first subtracting 5x from both sides of the inequality sign as shown below:
.
T + 5x - 5x - 2 > 6x - 5x + 7
.
On the left side the +5x and the -5x cancel each other. And on the right side, the 5x is subtracted from 6x to result in just x. After this, the inequality is reduced to:
.
T - 2 > x + 7
.
To get rid of the -2 on the left side, we just add 2 to both sides:
.
T - 2 + 2 > x + 7 + 2
.
On the left side the -2 and +2 cancel each other, and on the right side the +7 and +2 add to +9, making the inequality become:
.
T > x + 9
.
This is read as T is greater than x + 9 or equivalently as X + 9 is less than T as shown below:
.
x + 9 < T
.
So we have now developed the two possibilities:
.
T < 11x + 5 and x + 9 < T
.
And these can be combined and written as
.
x + 9 < T < 11x + 5
.
This means that answer (b) is the correct choice.
.
Hope this helps you to understand the problem and how you can work it to get the answer.
.

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If you are trying to solve a standard quadratic equation of the form:
.

.
and you notice that there is no term containing x, then you can use the quadratic formula if you just set b equal to zero. For the example problem you gave:
.

.
by comparing this to the standard quadratic equation, you can see that a = 4, b = 0, and c = -36. Then by the quadratic formula you can say that:
.

.
and substituting the values we identified for a, b, and c we get:
.

.
Notice that there is no "b" term before the radical in the numerator. Also there is no b^2 term in the radical. So the numerator is just the radical which is preceded by a + and - sign, and the radical contains just the product -4 times 4 times -36. And the denominator is 2*4 which is 8. So the answer is:
.

.
Doing the multiplication within the radical results in :
.

.
The square root of 576 is 24, so the answer becomes:
.

.
And dividing it out results in:
.

.
And don't forget the + and - signs to give you p = +3 and -3.
.
But there is even an easier way to do this example problem ... with fewer chances for error because there are less manipulations. Start with the problem as given:
.

.
Notice that you can divide both sides (all terms) by 4 to get:
.

.
Then get rid of the -9 on the left side by adding +9 to both sides as shown:
.

.
Then take the square root of both sides to get the answers:
.
and
.
Hope this helps you to understand the problem a little better.
.

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All these two problems are telling you to do is to replace the variable in the function with the quantity in the parentheses.
.
So for C(2) just replace g with 2.
.
Start with C(g) = 3.03(g)
.
For the first problem replace every g with 2. When you do that you get the equation:
.
C(2) = 3.03(2)
.
Multiply out the right side by multiplying 3.03 times 2 and the result is:
.
C(2) = 6.06 .... that's the first answer.
.
For the second problem, follow the same process, but replace every g with "a"
.
Again start with C(g) = 3.03(g)
.
Replace every g with a to get:
.
C(a) = 3.03(a)
.
And that's the answer to the second problem you were given.
.
Pretty easy once you understand what the problem is asking you to do.
.
Hope this helps.

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Don't get too lost trying to understand numbers. A lot of it is just relaxing and using your common sense. Let's try it on this problem.
.
First we recognize that we don't know how old Roberta and Rachel are at the present time. So, let's just say that currently Roberta is X years old and Rachel is Y years old.
.
From math we need to know one thing. If we have two unknowns (which X and Y are) then we will need two separate equations to find these two unknowns. That being the case, let's see what the problem tells us that will lead us into writing two equations.
.
First the problem tells us that Roberta (who is presently X years old) is three years younger than Rachel (who is presently Y years old). Since Roberta is the younger by three years, we can find Rachel's age by adding 3 years to Roberta's age and that answer will equal Rachel's age. In other words:
.
X (which is Roberta's age) plus 3 years equals Rachel's age (which is Y)
.
Or in shorter form: X + 3 = Y <----- this is one of the two equations
.
Next the problems talks about 8 years ago. What do we know about that? Presently one of the girls (Roberta) is X years old. How old would she have been 8 years ago? Think about how old you were 8 years ago. You would find that by taking your present age and subtracting 8 from it. Same thing for Roberta. We take her present age (X) and subtract 8 from it. In shortened form it is X - 8. Same thing for Rachel. Her present age is Y so 8 years ago she would have been Y - 8 years old.
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So eight years ago Roberta's age was X - 8 and Rachel's age was Y - 8. But Roberta was half of Rachel's age at that time. Suppose you had a friend and you were half her age. How would you find her age? You could double your age. Or if you knew her age, you could take half of it and get your age. Roberta is the younger, So we could double her age and get Rachel's age. Or we could take half of Rachel's age and know that it would equal Roberta's age. Either way ... it will make no difference.
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So we could double Roberta's age (X - 8) and it would equal Rachel's age (Y - 8). Or we could take half of Rachel's age and it would equal Roberta's age. In equation form these would be:
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Doubling Roberta's age to equal Rachel's age ... 2*(X - 8) = Y - 8 or
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Taking half of Rachel's age to equal Roberta's age ... (1/2)*(Y - 8) = X - 8
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We can use either one of these equations as the second equation needed to solve the problem. (If we worked on simplifying both of these equations, we would find that they turn out to be identical. They are just different ways of saying the same thing.)
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Let's use the first of these two equations ... the one doubling Roberta's age ... just because it doesn't contain a fraction to work with. The other equation (taking half of Rachel's age) contains that fraction 1/2 we would have to deal with. (But it would work out to give us the same results.)
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So our two equations are:
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X + 3 = Y and
2*(X - 8) = Y - 8
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Let's use the substitution method of solving these two. From the first equation we see that Y equals X + 3. So we can go to the second equation and in it replace Y by X + 3 to make it become:
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2*(X - 8) = X + 3 - 8
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Do the distributed multiplication on the left side by multiplying 2 times each of the terms in the parentheses. When we do that the left side becomes 2X - 16 and the equation then is:
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2X - 16 = X + 3 - 8
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On the right side combine the +3 and the -8 to get -5 and this reduces the equation to:
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2X - 16 = X - 5
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Now we can get rid of the -16 on the left side by adding 16 to both sides. When we add 16 to the left side, it combines with the -16 to cancel each other out. And when we add 16 to the right side it combines with the -5 to give +11. The effect is that the equation becomes:
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2X = X + 11
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Then we can get rid of the X on the right side by subtracting X from both sides. When we do that the equation reduces to:
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X = 11
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This tells us that X (Roberta's present age) is 11. Since Roberta is 3 years younger than Rachel, we then know (from our first equation) that Rachel must be 14. And 8 years ago, Roberta would have been 3 and Rachel would have been 6. So Rachel would have been double Roberta's age (or Roberta would have been half of Rachel's age).
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See how things work out?
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I hope this helps you to understand the problem and gives you a little insight into dealing with whether you double or take 1/2 of the ages 8 years ago. You can do it either way as long as you keep track of whose age you are doubling or whose age you are taking half of.

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Given to solve:
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7(x-9)+3x<-3(-4x-9)-3x
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You can generally solve these by following the same rules as you would for solving an equation EXCEPT that for solving inequalities, if you multiply or divide both sides by a negative quantity, you must also reverse the direction of the inequality sign. This being the case, let's begin:
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A good place to start is to do the distributed multiplications on both sides of the inequality sign. On the left side multiply the 7 times each of the two terms in the parentheses. And on the right side multiply the -3 times each of the two terms in the parentheses. After these multiplications the inequality becomes:
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7x - 63 + 3x < 12x + 27 - 3x
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On the left side add the 7x and +3x to get 10x. And on the right side add the 12x and the -3x to get 9x. As a result, the inequality is simplified to:
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10x - 63 < 9x + 27
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Get rid of the 9x on the right side by subtracting 9x from both sides to get:
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x - 63 < 27
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Finally, get rid of the -63 on the left side by adding 63 to both sides and you have the answer:
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x < 90
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This says that the original inequality will be true if you substitute for x any value that is less than 90. Let's just run a sample check. In the original inequality let's substitute 89 for x. Since x is then less than 90, the inequality should be correct. By substituting 89 for x the original inequality becomes:
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7(89 - 9) + 3*89 < -3(-4*89 - 9) - 3*89
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On the left side, in the parentheses the 89 - 9 becomes 80. And on the right side, within the parentheses the -4 times 89 is -356 and then subtracting 9 from that results in -365. So these two changes make the inequality become:
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7(80) + 3(89) <-3(-365) - 3*89
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On the left side the 7 times 80 is 560 and the 3 times 89 is 267. On the right side the -3 times - 365 is + 1095 and the -3 times 89 is - 267. Substituting these changes into the inequality simplifies it to:
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560 + 267 < 1095 - 267
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Now adding the 560 + 267 on the left side results in +827. And on the right side subtracting 267 from 1095 results in +828. This makes the inequality simplify to:
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+827 < +828
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This certainly is true. 827 is less than 828. So if x is 89, the inequality holds true.
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But what happens if we let x equal 90? This means that x equal to, but NOT LESS than 90 as our answer said it should be. When x is equal to 90, the original inequality becomes:
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7(90 - 9) + 3*90 < -3(-4*90 - 9) -3*90
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Let's combine the numbers inside the parentheses. On the left side they combine to 81 and on the right side they combine to -369. This makes the inequality:
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7*81 + 3*90 <-3(-369) - 3*90
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Doing the two multiplications on the left side gives 567 + 270. Doing the two multiplications on the right side results in 1107 - 270. So the inequality reduces to:
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567 + 270 < 1107 - 270
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Adding 567 + 270 on the left side results in 837. And on the right side subtracting 270 from 1107 also results in 837. So the inequality becomes:
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837 < 837
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But this inequality is NOT TRUE. 837 is not less than 837 ... the two sides are equal. Therefore, when x equals 90, the inequality DOES NOT WORK. By these two tests we know that when x equals 89 the inequality is OK, but when x equals 90 it is not. This suggests that x must be less than 90 as our answer told us. These two trials should give us confidence in our answer of x < 90.
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Hope this helps you to understand the problem a little better. Note that we did not use the rule that if you multiply or divide both sides of an inequality by a minus (or negative) quantity, you must also change the direction of the inequality sign. Remember that rule for another problem in which you may need to do such a multiplication or division.

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Your are at present 26 years old and your daughter is 4 years old.
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Some unknown number of years from now (call it x years) you will be 26 + x years old and your daughter will be 4 + x years old.
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At that point in time your age (26 + x) will be 3 times your daughter's age or 3 times (4 + x). Written in equation form, this equality is:
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26 + x = 3(4 + x)
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Do the distributed multiplication on the right side by multiplying 3 times each of the terms inside the parentheses. With that multiplication the equation becomes:
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26 + x = 12 + 3x
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To solve this equation, we collect the terms containing x by themselves on the left side, and the numbers or constants by themselves on the right side. Do this by subtracting 3x from both sides and also subtracting 26 from both sides. This subtraction is as shown below:
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26 - 26 + x - 3x = 12 - 26 + 3x - 3x
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On the left side of the equation the 26 - 26 cancel each other out and the x - 3x combines to be - 2x. The left side becomes as shown in the equation below:
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-2x = 12 - 26 - 3x
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On the right side the 12 - 26 combine to -14 and the 3x - 3x cancel out to zero. Therefore, the right side of the equation is as shown in the equation below:
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-2x = -14
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Solve for x by dividing both sides by -2. When you do that division the equation becomes:
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x = -14/-2
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and this simplifies to:
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x = 7
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This translates to telling you that 7 years from now you will be 33 years old (26 + 7) and your daughter will be 11 years old (4 + 7). At that point in time you will be 3 times older than your daughter (3 times 11 equals 33). So this is the answer to the problem. You will be 33 years old when you are 3 times your daughter's age.
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Hope this helps you to understand the problem and shows you a proces you can use to solve it.
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I presume that you are working with logarithms having the base 10.
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You are asked to solve for x in the equation:
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Collect the logarithmic terms on the left side of the equation by subtracting from both sides as follows:
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On the right side the two logarithmic terms are of opposite signs and therefore the cancel each other out. When you do that cancelling subtraction, the equation reduces to:
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By the rules of logarithms, the subtraction of two logarithms is equivalent to the logarithm of their quotient (in which the quantity in the subtracted logarithm becomes the divisor or denominator). This means that the left side of the preceding equation is equivalent to the left side shown below, and the right side of the equation remains as 1.
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Now convert this to the exponential form of a logarithm. Do that by raising the base (in this case 10) to the power of the right side of this logarithmic equation (in this problem that is the 1 on the right side) and set that result equal to the quantity the logarithm is operating on ( in this problem that is the quotient of (7x+1)/(x-2)). In equation form this becomes:
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Just to make this be a little more conventional in form, transpose this equation (switch the sides) to get:
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The right side has 10 raised to the first power, and that is just 10. So you can eliminate the exponent 1 and just write:
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Then get rid of the denominator on the left side by multiplying both sides of this equation by (x - 2) as follows:
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Cancel the common term in the numerator and the denominator on the left side:
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and you are left with:
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Do the distributed multiplication on the right side by multiplying 10 times each of the terms in the parentheses to get:
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Collect the terms having an x on the left side of the equation by subtracting 10x from both sides. Also collect the constants on the right side by subtracting 1 from both sides. This is done as follows:
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Combine the like terms on both sides. On the left side the 7x - 10x results in -3x and the +1 - 1 results in zero and need not be shown on the left side. On the right side the 10x - 10x results in zero and need not be shown and the -20 and -1 combine to -21. So the equation is simplified to:
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Solve for x by dividing both sides of this equation by -3 and you get:
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and this results in the answer:
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Hope this helps you to become a little more familiar with some of the rules you can use to solve logarithmic equations.
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You are asked to simplify:
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The power rule of exponents says that you can raise each of the factors within the parentheses by the exponent that applies to the set of parentheses. So in this case you can raise the factor 128 and the factor x^21 by the exponent 4/7 as follows:
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Now let's look at the 128. It is being raised to the 4/7 power. The numerator 4 means take the 4th power, and the denominator 7 means take the 7th root. We can do that in one of two ways. We could raise 128 to the 4th power and then take the 7th root of that quantity. Or we could take the 7th root of 128 and then raise that to the 4th power. Either way, the result will be the same.
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It will be easier to use the second method, because it is fairly easy to find the 7th root of 128 by trial and error. It is 2 because if you multiply 2 by itself 7 times the answer is 128. So we know that the 7th root of 128 is 2 and then we raise 2 to the 4th power and the answer (2*2*2*2) is 16. So 128 to the 4/7th is 16.
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In the same way we need to raise x^21 to the 4/7th. We can either raise x^21 to the 4th power and then find the 7th root of that, or we can find the 7th root of x^21 and raise that to the 4th power. This time it doesn't make much difference which one of the two methods we use. We can raise x^21 to the 4th power by multiplying 21 times 4. The answer is x^84. Then we find the 7th root of that by dividing 84 by 7 and we get x^12. Or we could divide 21 by 7 first to get x^3 and then raise that to the 4th power by multiplying 3 * 4 to get x^12.
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Now we have the information to get the answer. We just multiply the two answers we got when we raised 128 to the 4/7th and when we raised x^21 to the 4/7th. The multiplication of 16 and x^12 is the answer. So we can say:
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Hope this helps you to understand the problem a little better.
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To make things a little easier, let's first get rid of the denominators by multiplying both sides of the equations (all terms) by 8 to get:
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Where an 8 appears in the numerator, it cancels with the 8 in the denominator:
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and we are left with:
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Now we want to get y by itself on one side of the equation and everything else on the other side. To do that we can subtract w and z from both sides of the equation:
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On the right side the w and the minus w cancel each other and also the z and the minus z cancel each other, and we are left with just y. So the equation is reduced to:
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Just transpose this (switch sides) to get it into the conventional form:
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And that's the answer to this problem. We have solved for y in terms of the other variables in the equation.
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Hope this helps you to understand what you were asked to do and how you would go about doing it.
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Given to solve:
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r - (-7) = - 1 - 6
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Before we start, let's talk about the general strategy for solving this equation. Our overall goal will be to get the unknown number r on one side of the equal sign and get a number on the other. Usually, but not always, we'll get the r on the left side and the number on the right side. An example of the solution might be r = 10.
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Now lets work toward that goal. You can see that we already have r on the left side. So we need to do a couple of things. First, there is a number on the left side, and we'll need to remove that from the left side by moving it to the right side. When we do that we'll have several numbers on the right side and we'll need to combine them into a single number.
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So let's start with the given problem and let's see what we can do with it.
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r - (-7) = - 1 - 6
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Here's a rule to remember. If you have a set of parentheses preceded by a + sign you can remove the + sign and the parentheses without changing the signs of the terms in the parentheses. But if you have a set of parentheses preceded by a - sign, you can remove that - sign and the parentheses, but in doing so you need to change the signs of all the terms in the parentheses.
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Notice on the left side we have a set of parentheses preceded by a minus sign. We can erase that - sign and the parentheses, but we need to change the sign of the -7 that is inside the parentheses. So we take out the - sign in front of the parentheses, and also take out the two parentheses symbols ( and ), but in doing so we change the -7 to a +7. After that, the equation becomes:
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r + 7 = - 1 - 6
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Now let's move the + 7 on the left side to the right side. We do this by subtracting 7 from the left side. But the rules of equations say that whatever you do to one side of an equation, you must also do to the other side. Therefore, since we are going to subtract 7 from the left side, we must also subtract 7 from the right side. This will make our equation look like this:
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r + 7 - 7 = - 1 - 6 - 7
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On the left side the + 7 and the - 7 cancel each other out and the left side becomes just r as shown below:
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r = - 1 - 6 - 7
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Now all we have to do is to combine the three numbers on the right side. The - 1 and the - 6 combine to give - 7 as shown:
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r = - 7 - 7
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Finally, we combine the - 7 and the - 7 to get - 14 and we have solved the equation for r as:
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r = -14
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How can we check this work? We do it by going back to the original problem, substituting -14 for r, and determining if the left side equals the right side. So in checking we start with:
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r - (-7) = - 1 - 6
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We replace r with -14 to get:
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-14 - (-7) = -1 - 6
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Then we remove the - sign and the parentheses (while changing the -7 to a +7 to get:
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-14 + 7 = -1 - 6
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The -14 and the +7 combine to -7 so the equation becomes:
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-7 = -1 - 6
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And on the right side the -1 and the -6 combine to -7 so we have:
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-7 = -7
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Since both sides are equal, and we got there by setting r = -14, then we know that when r equals -14 the equation works. Therefore, we know the answer is correct.
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The answer to this problem is r = -14
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I hope this helps you to understand how to work problems such as this one.
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Given to solve for x
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Subtract log(x-6) from both sides to get:
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By the rules of logarithms, the subtraction of two logarithms is the same as the logarithm of their division with the divisor being the quantity in the negative logarithm. So we can writ that the above equation is equivalent to:
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Next convert this to exponential form by raising the base (base 10) of the logarithm to the power which is on the right side of the equal sign and setting this equal to the quantity that the logarithm operator is acting on. This results in:
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The 10 to the first power is just 10. So the equation is:
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Get rid of the denominator on the left side by multiplying both sides by x - 6:
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Do the distributed multiplication on the right side by multiplying 10 times each of the two terms in the parentheses, and the result is:
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Subtract 10x from both sides:
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Subtract 3 from both sides:
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Multiply both sides by -1:
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Solve for x by dividing both sides by 4 and you arrive at the solution:
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Hope this helps you to understand how this problem can be worked by using the properties of logarithms in a logical way.
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This is more of a logic puzzle than a math problem.
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Throw Switch #1 to the "ON" position, and leave it in that position for a couple of minutes. Then set Switch #1 back to the "OFF" position and quickly throw Switch #2 to the "ON" position. Immediately go upstairs and observe the bulb.
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If the light is on, Switch #2 controls the light because it is the only switch in the "ON" position.
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If the light is off then Switch #2 does not control the bulb. That being the case, feel the bulb. If the bulb is very warm to the touch, Switch #1 turned it on previously, so it (Switch #1) controls the bulb.
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But if the light is off, and it is cool to the touch, then neither Switch #1 nor Switch #2 turned the light on, so the light must be controlled by Switch #3.
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Your answer is correct.
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Because the first set of parentheses is preceded by an implied + sign, the parentheses can be removed without changing the terms inside. So after removing the first set of parentheses the problem simplifies to:
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2x^2 + 5x - 4 - (3x^2 - 7x + 1)
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In this simplified version of the problem, the set of parentheses is preceded by a - sign. Therefore, you can remove the parentheses if you also change the sign of every term within the parentheses. So when you remove the parentheses, the problem now simplifies to:
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2x^2 + 5x - 4 - 3x^2 + 7x - 1
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Now group all terms having like powers of x as follows:
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2x^2 - 3x^2 + 5x + 7x - 4 - 1
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and then combine each pair of the like terms and you have:
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-x^2 + 12x - 5
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This is exactly as you said it should be.
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Good job and keep up the good work. I hope this gives you confidence that you understand what to do with problems such as this one.
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Hi David --
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I think I understand the problem from your description, so I'm going to try to help. If I have misunderstood the problem, please post it again and one of the tutors will probably help.
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I'm going to say that phasor PQ has its tail (P) at the origin and extends to the right along the positive x-axis for 20 units (20A) to its head (Q). At this point the phasor QR has its tail and it goes up and to the right at 35 degrees to its head at point R. Its length is 14A. If this is correct, make a sketch of it. (A picture is worth a thousand words of explanation.)
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Now on your sketch, go to point R and drop a vertical line down so that it intersects the x-axis at a point we will call point S. The next thing we are going to do is to break phasor QR into a horizontal component QS and a vertical component SR. (Note that phasor QS will have its tail at Q and its head at S, and phasor SR will have its tail at S and its head at R. Also, note that angle QSR is a 90 degree angle so we have formed a right triangle QSR in which angle RQS is the 35 degree angle. Phasor QS is the adjacent side of the 35 degree angle, phasor QR is the hypotenuse, and phasor SR is the opposite side.
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Using the definition of cosine as being the adjacent side divided by the hypotenuse, we can now write:
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Then we can multiply both sides by QR (and transpose or switch sides) to get:
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And we know that QR is 14 units long, so we can substitute 14 for QR to make the equation become:
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Do the multiplication on the right side of this equation and you should get (to three decimal places) that:
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We can do a similar analyses for the length of SR by using the definition of sine (side opposite/hypotenuse = SR/QR) to get:
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Do the multiplication on the right side and you get the answer the length of SR as being:
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Now we can say that we have formed a new phasor PS that is the sum of phasor PQ and phasor QS. It, therefore, is 20A + 11.468A = 31.468A long. And we also have the perpendicular phasor SR that is 8.030A in length. But observe that phasors PS and SR add together to form phasor PR (tail at P and head at R) and we are trying to find the magnitude (or length) of phasor PR and the angle it makes with positive x-axis.
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We can use trigonometry to find the angle that phasor PR makes with the positive x-axis - the angle RPS) and also the magnitude of phasor PR. We know that in triangle PSR the side opposite to the angle RPS is phasor SR and the side adjacent is phasor PS. With this information we can use the definition of tangent as side opposite divided by side adjacent. Therefore for angle RPS we can write:
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Then substitute 8.030 for SR and 31.468 for PS and this equation becomes:
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Doing the division on the right side results in:
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Rounded to three decimal places. So now all we have to do is find the arctan (or if you prefer, tan^(-1)) of 0.255. Using a calculator and rounding the resulting angle to 3 decimal places we find that the angle RPS is 14.306 degrees. That's one of the answers that we were looking for.
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Now we need to find the magnitude of phasor PR. We could use the Pythagorean theorem because triangle RPS is a right triangle with RP as the hypotenuse, PS as one leg, and SR as the other leg. We could write the Pythagorean theorem as:
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Then substituting 31.468 for PS and 8.030 for SR we get:
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Squaring the two terms on the right, adding those results, and taking the square root of the sum results in finding the magnitude of phasor PR as:
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We can find the magnitude of PR by using either the sine or cosine function. We know that the angle that PR makes with the x-axis is 14.306 degrees and that the side opposite this angle (SR) has a length of 8.030. So we can write the sine function as:
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The sine of 14.306 degrees is 0.247. Substitute that into this equation and you get:
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Multiply both sides by PR and then divide both sides by 0.247 and the equation becomes:
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Do the division on the right side and you get the answer for the length of PR as being:
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and this is fairly close to what we got using the Pythagorean theorem. We could likely decrease the disparity by carrying more decimal places for all our calculations, but that makes a lot more typing necessary.
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Any how, to summarize our results phasor PR has a magnitude of around 32.5A and makes an angle with the x-axis of about 14.3 degrees.
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Please check all this work and analysis. You should probably carry more decimal places in all your calculator results. I haven't had my morning coffee jolt yet, so there are no guarantees about any of this work, but hopefully this discussion is enough to give you the idea of how to proceed with getting the answer to this problem.

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In the first problem, you are asked to solve:
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The general plan will be to get the numbers on the right side of the equal sign and combine them, and get just x by itself on the left side.
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So first, let's get rid of the -3 on the left side. We do this by adding +3 to the left side. But if we add +3 to the left side, we must also add +3 to the right side to keep the equation in balance. Adding +3 to both sides results in:
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On the left side the -3 and the +3 cancel each other because they total zero. And on the right side the -1 and the +3 add to give +2. So by adding +3 to both sides the equation becomes:
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Finally, to get rid of the 6 in the denominator of the left side, we multiply the left side by 6. When we do that, we must also multiply the right side by 6 to keep the equation in balance. So first let's do the multiplication of both sides by 6 as shown below:
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On the left side the 6 in the numerator cancels with the 6 in the denominator as shown and on the right side the 6 times the 2 results in 12:
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And we are left with the answer:
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You had the correct answer, but I'm not exactly sure how you went about solving it. And since you were unsure of the answer, I expect that you were also unsure of what you did. It looks as if you correctly added the +3 to both sides, but you then appear to have divided both sides by 2 and that would not be a correct thing to do. The above explanation should help you see what you can do and why you do it.
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Next you were given:
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The plan for this is to work it as you would an equation, but instead of an equal sign you have an inequality sign. In general you can use the same procedures as you would for an equation, except that if you multiply or divide both sides by a negative number, you also reverse the direction of the inequality sign.
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It appears as if you decided to add +5 to both sides of this inequality sign. That was correct. So let's do it here also:
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On the left side of the inequality sign the -5 and the +5 cancel each other. And on the right side the 13 and the +5 add together to give 18. The result is that when we add +5 to both sides we get:
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It now appears as if you decided to divide one side by 8 and the other side by 8x. This is not quite correct. The thing to do next is remember that we are trying to get all the numbers together on the right side and all the x terms combined on the left side. So let's get rid of the 2x on the right side by subtracting 2x from both sides. When we do that the inequality becomes:
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On the left side the 8x and the -2x combine to give 6x. And on the right side the +2x and the -2x cancel each other. So we are left with:
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Finally, we can get rid of the 6 on the left side by dividing both sides of this inequality by +6 as shown below:
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The six in the numerator of the left side cancels with the 6 in the denominator. And on the right side the 6 in the denominator divides into the 18 to give 3. So dividing both sides by 6 results in:
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Which simplifies to:
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This says that x is greater than 3. What this means is that the original inequality you were given will be true if x is a number greater than 3. Let's try it for a number greater than 3. Let's say that x is +4 and then let's substitute that value for x in the original inequality. The original inequality is:
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Substitute +4 for x and you have:
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On the left side the 8 times 4 multiplies out to 32 and when you subtract 5 from that the left side reduces to 27. On the right side the 2 times 4 multiplies out to 8 and when you add that to 13 you get 21. So with x equal to +4 the original inequality becomes:
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and this says that with x equal to 4 (which is greater than 3) the inequality tells us that 27 is greater than 21. This is correct and it gives us some assurance that if x is greater than 3, the inequality is likely correct.
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Let's try the same sort of analysis using a value for x less than 3. Zero is less than 3 and it is easy to work with in this case. So let's set x equal to zero in the original inequality. When we substitute zero for x in the original inequality it becomes:
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The two terms that are products involving zero disappear and the inequality simplifies to:
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But this is not true. -5 is not greater than 13. So for one value of x that is less than 3 we know the inequality does not hold. This also helps us to say that x must be greater than 3 in order for the inequality to hold true.
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So the answer to this second problem is that in order for the inequality to be true, x > 3 ... x must be greater than 3.
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I hope that the above discussion of these two problems help you to understand how you can work through them to get a correct answer.
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You can put this solution on YOUR website!
given to solve for x:
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First, take the base 10 log of both sides:
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Using the product rule for logarithms, split the left side into two logarithms:
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Subtract log(6) from both sides to get:
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By the rules for logarithms, an exponent can be brought outside as a multiplier. Therefore, on the left side, bring the x out as a multiplier of the logarithm. The result is:
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Divide both sides by log(4):
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Use a calculator to find that log(99) = 1.995635195, log(6) = 0.77815125, and log(4) = 0.602059991. Substitute these values into the equation for x to get:
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Subtract the two terms on the numerator:
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Do the division on the right side and you have:
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The problem tells you to round this answer to two decimal places. When you do that, the answer shortens to:
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That's the answer that you're looking for. Hope this helps you to understand the process and the rules that apply to logarithms better.
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You can put this solution on YOUR website!
In the work that follows the software that renders equations to look like you would write them. The software has some troubles with handling fractional exponents and radical signs in exponents. In the case of radical signs, they appear to look more like just a check mark because they are lacking the horizontal line above the quantity that is within the radical. You'll have to make a corrective adjustment in picturing the way it should be. Sorry about that. In other situations fractional exponents get clipped off. I'll note where this happens so that you can understand what the exponent should look like. Again, my apologies.
The book answer shows that two values of x will make the left side of this equation equal to the right side:
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<--- note that the top of the radical sign is likely missing
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The first solution (x = 0)is intuitively obvious. It involves the fact that if any number is raised to the exponent zero, the answer will be 1. Note that both exponents in this equation will be zero if x is set equal to zero. With x equal to zero the equation becomes:
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<---- again, the top bar of the radical sign is likely gone
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Since the square root of zero is zero, the left side of this equation becomes as shown below:
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and with both exponents now 0 the equation becomes:
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This obviously is true, so when x equals zero, the equation is true. Therefore, x = 0 is one solution to the problem.
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Finding the other solution is a little less obvious. Start with the given equation:
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<---- again the top bar of the radical is likely missing
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Now instead of writing the exponent of 9 with a radical sign, let's change the exponent so that x has an exponent of 1/2 itself. An exponent of 1/2 is equivalent to the square root radical. That being the case we can write the equation in the equivalent form:
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<---- the 1 in the numerator of the exponent 1/2 is likely clipped off a little
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But we know that 9 is equal to 3 squared (or ). Substitute this for 9 and the equation becomes:
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Now using the power rule of exponents, we can multiply the two exponents (2 and x^(1/2)) on the left side to make the equation become:
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<---- on the left side the x in the exponent has an exponent of 1/2. It is clipped so that only some of the 2 is visible. Think of it as 1/2
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Note now that the bases on both sides of this equation are 3. Therefore, for the sides to be equal, the exponents have to be equal. In other words we can set the exponents on both sides equal to each other to get the equation:
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<--- the numerator 1 is likely to be slightly clipped off
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Square both sides of this equation and you get:
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Divide both sides by x and the equation then becomes:
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This tells you that both sides of the original equation become equal if x equals 4. You can check this by returning to the original equation:
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<---- the radical sign is missing the top horizontal line
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Substitute 4 for x and you get:
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<---- and again the radical sign is missing its top
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But the square root of 4 is 2 so the equation becomes:
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Square 9 and you get 81 on the left side. Then if you raise 3 to the fourth power you also get 81 on the right side. This means that the answer x = 4 satisfies the equation by making both sides equal and therefore x = 4 is a valid solution.
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Hope this helps you to see how to get the two answers to this problem.
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You can put this solution on YOUR website!
I think that you intended for the exponent in the denominator to be 16a rather than the way you wrote it. In other words you meant:
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This being the case, then the next step would be to replace the square root radical with the exponent 1/2, which converts the problem to:
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Following the power rule for exponents in the term in the denominator we can multiply the exponent (1/2) times the exponent (16a) to get an answer of 8a. The problem can therefore be rewritten as:
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Then, since the base for the exponential terms is x in both the numerator and the denominator, we can divide by raising the base to the difference between the two exponents as follows:
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Subtracting the exponents results in the answer you thought it was, namely:
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I hope that my interpretation of the problem you wanted help with is correct. If not, please post it again and one of the tutors will likely respond.
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