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(15 – 5) ÷ [(12 ÷ 2 · 2) – 2]
.
First do the work inside the parentheses. The (15-5) becomes just 10.
.
Then go to the interior set of parentheses in the next expression. That is the parentheses
that contain (12 ÷ 2 · 2). First do the multiplication and divisions from left to right. That means
the first thing to do is divide 12 by 2 to get 6. Then multiply that by 2 to get 12. So you
can replace (12 ÷ 2 · 2) by 12. [There were no additions or subtractions in these parentheses
so you didn't need to do any adds or take aways next from left to right.
.
At this point the original problem has been simplified to:
.
10 ÷ [12 - 2]
.
Do the work in the brackets (which are a different way of showing parentheses) and you
get that 12 - 2 = 10. The equation is now reduced to 10 divided by 10 and the answer to
that is obviously +1.
.
The rules for working in single line problems are:
.
First do the work inside parentheses
Second work on any terms that have exponents
Third do the multiplications and divisions in order from left to right
Fourth do the additions and subtractions in order from left to right
.
[Note that in working inside parentheses you may have to use the second, third, and fourth
rules before going on to the next set of parentheses.]
.
Easier to say than it is to recognize what to do next, isn't it? It sure would have been
easier to see if the original problem had been written:
.

.
But that's just my opinion ...

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.
This is in the form of a proportion ... two fractions set equal to each other. There are
ways that you can work through common denominators and do simplifications, but a common
method is that a lot of students use for solving proportions that aren't way complex is
to begin by cross-multiplying them ... multiplying the denominator on one side by the numerator
on the other side and setting the two products equal. For this problem the would
multiply the and the would multiply the . You can do
that multiplication in your head and get the resulting equation:
.

.
Next do the multiplications on both sides to get:
.

.
The rest is straightforward. Add +140 to both sides to eliminate the 140 on the left side:
.

.
Then add -54x to both sides to eliminate the 54x on the right side:
.

.
Finally, to solve for x, just divide both sides by -26 and the answer becomes:
.

.
That's the answer. If you do the division you get a decimal answer of .
.
Hope this helps you a little to see a way of working problems such as these that are in
proportional form. This method is not the only way the problem could be done, but it's
a convenient way at times.

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(3,_),(_,-1),2x-3y=5
.
This is two separate problems. The first problem is that you are given (3,_) and the equation
.
2x - 3y = 5
.
The 3 in (3,___) represents x. Substitute 3 for x in the equation and solve for y:
.
2(3) - 3y = 5
.
Multiply out the left side to get:
.
6 - 3y = 5
.
Subtract 6 from both sides:
.
-3y = -1
.
Divide both sides by -3 to end up with:
.
y = (-1/-3) = 1/3
.
So the missing entry in (3,_) is 1/3 and the answer becomes (3, 1/3)
.
The second problem involves the answer set (_, -1) for the equation 2x - 3y = 5. Substitute
-1 for y in the equation to get:
.
2x - 3(-1) = 5
.
And multiply out to get:
.
2x + 3 = 5
.
Subtract 3 from both sides:
.
2x = 2
.
Divide both sides by 2 and the result is:
.
x = 1
.
So the missing x value in (_, -1) is 1 and the answer set is (1, -1)
.
Hope this helps you to understand what the problem was looking for.

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Given
.
The slope intercept form of an equation is:
.

.
where m represents the slope and b is the value of y where the graph crosses the y-axis.
.
So to put the given equation into the slope intercept form, you need to have just y on the left
side and the x term plus a constant (that could be zero) on the right side. The first thing
that is evident is that you need to get rid of the -3x on the left side. Do this by adding
+3x to both sides. The result is:
.

.
Note that this is exactly like the slope intercept form. m is the 3 and b is the +6.
So you can put a dot on the y-axis at +6 because that is b and b is where the graph crosses
the y axis. Next, the slope of +3 means that the graph rises 3 units for every 1 unit that
you move horizontally to the right. You can draw this by putting your pencil on the dot you
put on the y-axis at +6. Then move your pencil one horizontal unit to the right. Stop. Now move your
pencil vertically up 3 units. Stop. You should be at the point (1,9). Put a mark at that
point because it is on the graph. From that point move horizontally 1 unit to the right.
Stop your pencil. Then move vertically up 3 units. Stop your pencil and mark the point. You
should be at (2,12). You now have 3 points on the graph. Put a straight edge along those points
and extend a line through them in both directions. That is the graph of the equation.
.
The reason you moved up after moving one unit to the right was because the slope was positive.
If the slope had been negative, then for every unit you moved your pencil horizontally
to the right you would then move vertically down a number of units equal to the slope.
.
Hope this helps you understand the slope intercept form a little better.

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.
Add -16 to both sides to solve the inequality for x:
.

.
This means that x is not allowed to be -6 (because it is greater than -6) and x can be
any value on the number line that is to the right of -6.
.
For example, let x = 0. The inequality of the original problem becomes:
.

.
Which simplifies to:
.

.
This inequality is correct ... 16 is greater than 10, so x can be 0
.
Hope this helps you to understand how to work the inequality given in this problem. Graph
it by excluding -6 and drawing the graph all the way to the right of -6 on the number line.

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.
Operate on this just as you would on an equation. For the first step, you can multiply
out the left side to get:
.

.
Combine the +5 and the +4 on the left side. The inequality becomes:
.

.
Eliminate the +9 on the left side by adding -9 to both sides:
.

.
And add the +6 and -9 on the right side:
.

.
Get rid of the 8x on the right side by adding -8x to both sides. The inequality simplifies to:
.

.
Dividing both sides by 2 results in:
.

.
This tells you that x can be anywhere on the number line below the value .
But x cannot equal and it cannot be any value to the right of
on the number line.
.
It doesn't happen in this problem, but remember this rule: if you multiply or divide an
inequality by a negative number you must change the inequality sign to the opposite
direction.
.
Hope this helps you see how to work with inequalities.

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.
Start by treating this just as you would an equation and solve for x. First multiply out
the left side. When you do the result is:
.

.
Next on the right side multiply out the -2 times (x+5). When you do the inequality
becomes:
.

.
Then on the right side combine the x and -2x to get just -x. The inequality is then:
.

.
Finally multiply out the right side:
.

.
To cancel out the +6 on the left side, add a negative 6 to both sides. The inequality is
then"
.

.
And to cancel out the -5x on the right side, add +5x to both sides. The result is:
.

.
To solve this for positive x, divide both sides by -4. But now you have to remember the
rule ... whenever you divide or multiply both sides of an inequality by a negative number,
the inequality reverses direction. Dividing by -4 and reversing the inequality gives you:
.

.
This tells you that when x is less than 14 the inequality of the original problem is
satisfied. And when x is 14 or more, the inequality of the original problem will not work.
.
Try a few values of x less than 14 and see if they don't work. Try x=14 and see why it doesn't
work. And try a value or two for x greater than 14 to verify that they don't work.
.
Note 0 is less that 14, so you can try it. It makes all the x's disappear and the problem
is simplified because of that.
.
Hope this helps you understand inequalities a little better. And don't forget the rule
about multiplying or dividing inequalities by a negative number.

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This is a "trick" question, so you need to be a little careful with it. It usually is
helpful to make a rough sketch of the x and y-axes and then plot the points, just to help
visualize the slope of the line. Then we could proceed to use the slope-intercept form of an
equation ... y = mx + b ... where m is the slope and b is the point at which the graph crosses
the y-axis.
.
But in this case something unusual happens when you try to calculate the slope.
.
For the two given points (5,5) and (4,5) you can identify the first point as (x1,y1).
And you can identify the second point as (x2,y2). By comparing each given point to its
corresponding form you can see that: x1=5, y1=5, x2=4, and y2=5. Now write the equation for
finding the slope (m) if you know two points on the line. This equation is:
.

.
Now plug in the values for x1, y1, x2, and y2. When you do the slope equation becomes:
.

.
But look at the numerator. 5 minus 5 is zero. The denominator equals -1. And when you
divide -1 into zero, the answer is zero. What does a slope equal to zero mean? If a line
has a slope of zero, it does not go upward or downward. It just stays horizontal.
.
So the graph is a horizontal line. It has the same value for y regardless of the value
for x. If you think about it a little bit, for this problem you will see that the line
goes across the y-axis at +5 and it runs in both directions as far as you want. Whatever
value you chose for x, y is always +5 units above the x-axis. And the equation for this
situation is y = +5. That's your answer ... choose any value on the x-axis and the corresponding
value of y is always +5.
.
Think about it a little bit and it will make sense. Also your sketch would have shown you
that the line was probably horizontal and then you could have verified that by using the
slope equation.

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The answer is -20 because the when you take the absolute value of any number becomes a
positive number. In this problem taking the absolute value of -20 makes it + 20, and that
is the largest number in the set.
.

.

.

.

.

.

.
Another way to look at this is that when you take the absolute value of a number you are finding
the distance that number is from zero on the number line. The absolute value of -5 is the
same as the absolute value of +5 because both numbers are 5 units away from 0 on the number
line. In your problem -20 is 20 units from zero on the number line.
.
And another way to look at it is to just convert all the numbers to positive values and
then choose the largest of these values as the answer to your problem.
.
Hope this helps you to visualize what the absolute value means regarding numbers. It
just leaves positive numbers as positive, and it converts negative numbers to positive
numbers.

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.
read this as 2.5 quarts per minute times 1 gallon per 4 quarts times 60 minutes per hour.
.
If you multiply the numbers in the numerators and then multiply the numbers in the denominators
you have:
.

.
Note how the dimensions work. The qts in the numerator cancel with the qts in the denominator,
and the min in the numerator cancel with the min in the denominator. All you are left with
is gal in the numerator and hr in the denominator. Therefore, the answer is 37.5 gallons per hour.

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This problem is the same as problem 70524. It was answered by answer 50310. Check that out.

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.
One method of solving this involves letting the entire quantity within the absolute
value signs first have a positive sign and then have a negative sign. Solve these two cases.
Perhaps it is best explained by using this problem as a typical example.
.
The quantity inside the absolute value signs is . First write the inequality
equation using +(x+2) for the left side. In this case the inequality becomes.
.

.
Solve this for x just as you would an ordinary equation ... by subtracting +2 from both
sides to get the answer as .
.
But that's only the first part of the solution. The second part says to give the entire
quantity inside the absolute value signs a negative sign and solve the inequality again.
.
So this time the inequality becomes:
.

.
You can remove the parentheses preceded by a negative sign by changing the sign of all the
terms inside the parentheses and then just erasing the parentheses. If you do this you
get:
.

.
Begin the solution by adding 2 to both sides to eliminate the -2 on the left side. When
you do that, the inequality becomes:
.

.
Next comes a tricky part. You want to solve for positive x, so you are going to multiply both
sides by -1. However, if you multiply or divide both sides of an inequality by a negative
number you must reverse the direction of the inequality sign. No big deal, just something to
remember. So in this case after the multiplication and sign reversal the solution for x
becomes
.

.
In summary, the regions on the number line that satisfy the original inequality are any
value of x equal to or to the left of -6 and any value of x equal to or to the right of +2.
.
You can do a quick check by returning to the original problem and solve it for x = -7,
and x = +3. Or use any convenient value in the two regions ... for example x = - 10 and
x = + 10. You should see that the inequality is satisfied. Plus you can use any value
of x between -6 and +2 to prove to yourself that the original inequality is not satisfied by
values of x in that region. Usually x = 0 works well for this check because all the terms
containing x just disappear. It's usually a good idea to check the points you found on
the number line also. If you let x = -6 and then let x = +2, you should find that in
both cases the inequality of this problem is true only because of the equal sign associated
with the less than or greater than sign. Therefore, for this inequality x = -6
and x = +2
are included in the solution set.
.
In summary, solve two problems ... first use the entire quantity inside the absolute value
signs with a plus sign and solve the inequality for +x. ... next use the entire quantity inside
the absolute value signs with a negative sign and solve the inequality for +x. Don't forget
multiplication (or division) of both sides of the inequality by a negative number,
requires you to reverse the inequality sign. Finally use numbers in each of the regions that
you identify for the answer, and plug them into the original problem just to make sure
they work correctly. It also pays to check the end point numbers (in this problem they
were -6 and +2) to validate what region they fall into (are they part of the solution set
or are they not). [In this problem they were in the solution set because of the equal sign.]
.
Hope you find this method easy to remember and useful to you.

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If the log form is log b (d) = c
.
(read that as log to the base b of d equals c)
.
then the corresponding exponential form is
.
You are given log 3 (27) = 3. By comparing this with the log form (see top line above) you can
see that b = 3, d = 27, and c = 3. Putting these numbers in the appropriate places in the
exponential form results in:
.
Note that this equation is true because if you cube 3 you get 27
.
You are also given log 125 (25) = 2/3. By again comparing this with the log form in the first
line of this reply, you see that b = 125, d = 25, and c = 2/3. Substituting these numbers
for their corresponding letters in the exponential form results in:
.

.
You may recognize that the exponent means that you can either square the number and
then take the cube root, or you can take the cube root and then square that number.
In the case of 125, it is easier to first take the cube root because it is 5, and then
you square that to get 25. This means the exponential form is true because the left side
of that equation equals the right side.
.
If you work a number of these, you will see the pattern between the log and exponential
forms and the translation back and forth between the two forms will become easier for you.
Hope this helps you see your way through this type of problem.

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Temporarily replace the inequality sign with an equal sign ... just because you are probably
more familiar with working on equations. This temporary maneuver results in:
.

.
This is a typical quadratic equation. This particular equation can be factored so that the
equation can be broken down to:
.

.
This equation will be true if either of the factors on the left side is zero, because that
would make the left side of the equation equal 0 ... the same value that is on the right side.
So, one at a time set the two factors equal to zero and solve for x:
.
First and by adding 3 to both sides this becomes
.
Then and by subtracting 1 from both sides this becomes
.
This tells us that for this problem you have critical points on the number line at
and . Draw a horizontal number line and mark the points +3 and -1 on that line.
There are three important regions on this number line ... the region extending to the left
from the point -1, the region between the points -1 and +3, and the region extending to
the right from the point x = +3. You need to examine just these three regions to tell which of
them satisfy the inequality equation.
.
Return now to the original inequality equation as given in the problem statement:
.

.
Now pick any convenient number that is in the first region to the left of x=-1. Suppose that
you choose x = -10. Plug that value in for x in the inequality equation and simplify the
results. The substitution of -10 leads to:
.

.

.
You can already see that this equation is true because 117 > 0. Now you know that numbers
in the region from -1 all the way down to negative infinity (but not including -1) will satisfy
the inequality equation.
.
Next examine the region with -1 on one end and +3 on the other end. Select a convenient value
of x between these two end points and see what happens to the inequality equation. A
convenient value is x = 0 because when it is substituted in for x, it cause terms that
have an x in them to become zero. The substitution is:
.
and this simplifies to: .
.
This is obviously not true. So all the points from -1 to +3 will not make the inequality
equation true. Therefore, x cannot be from -1 to +3 including the end points. (Plug -1
and +3 into the inequality equation and see if the result is GREATER than zero.)
.
Finally do the same exercise for a single convenient value in the region where x is greater
than +3. It might be convenient to use x = +10, but you can use any value greater
than +3. If you do this you will find that it does satisfy the inequality equation,
so all values in that region of x greater than +3 all the way out to + infinity will satisfy
the equation.
.
In summary the inequality equation is satisfied if x < -1 or x > +3.
.
Hopefully this refreshes your memory on problems such as these.

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Start with
.
The fact that the point (3,a) satisfies the system of equations (that means it satisfies both equations), tells you that when x = 3 and y = a, the equation
is satisfied. So plug in 3 for x and a for y. When you do, the equation becomes:
.

.
This simplifies to:
.

.
Subtract 12 from each side of this equation to get:
.

.
Finally, divide both sides by 5 to get:
.

.
So now you know that a = -2 and that is one of the things you had to find. You can substitute
this into the point (3,a) which is known to work in both equations because it is the solution
to the system of equations. By substituting -2 for a you know that the point (3, -2) satisfies
both equations.
.
The second equation of the system of equations is:
.

.
Since you know this equation is satisfied by x = 3 , y = -2 you substitute these values into
the equation and both sides should be equal. By substitution the equation becomes:
.

.
Multiplying out the terms on the left side results in:
.

.
and adding the two terms on the left side tells you that b = 22. That is the second
quantity that you were supposed to find.
.
Hope this helps you to see your way through this word problem OK.

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.
You cannot add or subtract fractions unless they have a common denominator. In this problem
we can use a common denominator of 30 because all the denominators in the problem are factors
of 30. There are 3 terms in this problem and we need to get each term so it has a denominator
of 30.
.
The first term is . We could multiply this fraction by
and it would not change the value of the fraction because equals 1, so we are
multiplying the original fraction by 1. When we multiply the fraction by , the
result is and this simplifies to .
.
The second term is and to get the denominator to equal 30, we multiply the fraction
by . The result of this multiplication is that the denominator becomes 30,
and the numerator becomes . So the term is now converted to
.
The last term is . To give this a denominator of 30 we multiply the fraction
by . The denominator becomes 30 and the numerator becomes
which simplifies to . So the fraction is converted to .
.
Substituting these three results into the original equation converts the problem to:
.

.
Since the denominator 30 is common to all terms we can eliminate it by multiplying
all the terms in the equation by 30. The result is that the multiplier of 30 cancels with
the denominator of 30 in all terms and the problem now involves dealing with only the
numerators as follows:
.

.
Combine the -50 and the - 45 on the left side to get:
.

.
Add 95 to both sides to cancel the -95 on the left side. This results in:
.

.
Subtract 6x from both sides to eliminate it on the right side.
.

.
And finally, divide both sides by 14 to get:
.

.
That's the answer to the problem. Hope this helps you to see a way of solving fractional
problems such as these.

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.
If x = 13, all you do is substitute 13 for x in the equation to get:
.

.
Do the multiplication of and you get an answer of . Substitute
this value into the equation in place of the multiplication to get:
.

.
Then do the addition on the right side and you find that the employee makes an hourly wage
of:
.
or $14.15 per hour.
.
Hope this helps you see how equations can work for you to solve problems.

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.
You deal with this problem in two different parts. Part 1 consists of finding .
When you do that division you get for the answer. So now your problem is of the
form:
.

.
Now for Part 2.
.
When you multiply you add the exponents to get an answer of
and when you divide you get an answer of . You problem calls
for you to divide so you need to subtract the exponent in the denominator
from the exponent in the numerator. So the answer to this problem is
which simplifies to
.
So for the complete answer you need to multiply the answer for Part 1 times the answer for
Part 2. The result is
.
One final step. Scientific notation generally calls for one number to the left of the decimal
point. In this case you have no number to the left of the decimal point (zero doesn't count).
So what you do is take away one ten from the exponent of 10, leaving and use that
10 to multiply the 0.4 ... making it become 4.0. So the final answer in scientific
notation is

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Please check your numerator and re-post your problem. I think you may have made a mistake
in your terms. Thanks

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The most difficult part of this problem is defining the length of one of the equal sides of
the isosceles triangle. Call one of the equal sides L and the base B.
.
The problem says that the leg L is 4 times the base B less 8 cm. In equation form you can
write:
.

.
The perimeter of the isosceles triangle is . But you now know that .
So, you can substitute for L in the equation for the perimeter to get:
.
for the perimeter. This can be simplified by adding all the terms
containing B and also adding the two terms of -8. When you do, you end up with:
.

.
for the perimeter. But the problem tells you that the perimeter is 29 cm. Therefore, you
can set the equal to to get the equation:
.

.
To eliminate the -16 term on the left side, add +16 to both sides. This results in:
.

.
Then divide both sides by 9 to find that
.
You need the answer for the length of one of the legs L. You already found that this length
is determined by the equation . All you then need to do is substitute
5 for B and calculate the right side of the equation for L. You should get the answer that
L = 12. Work it out to make sure that is the correct answer. And you can check the problem
by adding L plus another L plus the length B to see if you get 29 for the answer.
.
Don't forget that you are working in cm so the values for L, B, and the perimeter
all have units of cm.
.
Also you need to think through how we got the equation that from the
wording of the problem.
.
Hope this helps you to develop skills at analyzing problems such as these word problems.

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From the way this problem is worded, the equation for C(x) is not needed to answer the questions
that are asked. The only equation that applies is:
.

The first question ... What is the profit if she sells 2000 units? Calculate the profit by
setting x equal to 2000 units so that the equation becomes:
.

.
The negative sign tells you that overall she loses $2500 if she sells only 2000 units.
.
The second question ... What is the profit if she sells 5000 units? Do the same calculation
as above, but this time set x equal to 5000 units so that the equation becomes:
.

.
The positive answer tells you that overall she makes $4,250 if she sells 5000 units.
.
So somewhere between sales of 2000 units (a loss) and selling 5000 units (a gain) she
reach a break-even point. At that break-even point her profit would be zero (neither a
loss nor a gain). So this time set the profit equal to zero in the equation and solve the
resulting equation for x.
.

.
Add 7000 to both sides and transpose the equation:
.

.
Divide both sides by 2.25 to get:
.
}
.
This result tells you that if she sells less than this amount she loses money, but if
she sells more than this number of units she makes money. Therefore, you can say that
she loses a very small amount of money if she sells 3111 units, and she makes a very small
amount of money if she sells 3112 units.
.
Hope this helps you to see your way through the problem ... especially with regarding
the information on the costs involved with production, C(x), and the sales price of $4.00
per unit. That information was not needed because you were provided with an
equation that
directly calculates profit based on the number of units sold.

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Given
.
To rewrite this as a function of x, you need to do two things ... first solve for y and then
replace y by f(x).
.
First solve for y. You need to move the -3x to the other side of the equation. You can
do that by adding 3x to both sides. On the left side the +3x cancels the -3x, and on the
right side the +3x appears. You now have:
.

.
You can now solve for just +y by dividing both sides by the coefficient (or multiplier)
of y. In this case the coefficient of y is +4, so divide both sides of the equation
by +4. The result is:
.

.
This simplifies to:
.

.
As the final step you replace y by f(x) to end up with:
.

.
and this has re-written the original equation (that was in standard form) into the form
of a function of x (denoted by f(x))
.
Hope this helps you to see how to do problems of this sort. Just follow the two-step
process ... first solve in for y in terms of x and then replace y by the notation f(x).

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Below is what I ended up with:
.
= -9x^2+8x^2+7x^2-5x^2-9x-6x-4x-3x+4+2+6-1 <----This is correct!
.
= x^2-4x+11 (<---is that right. No it's not. Look at -4x. How did you get the -4? The x^2 and
the +11 are OK.)
.
You obviously know how to work these, but just made a minor mistake. Keep up the good work!

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If f(x)=5x-1, find the following:
f(a-2)
Look in the sets of parentheses that follow the "f". The second set of parentheses
contains the expression a-2. The first set of parentheses just contains x. So the problem
means that there is a correspondence between x and a-2. All you do is go to the expression
.
f(x) = 5x - 1
.
and replace every x by a-2. When you do, the expression becomes:
.
f(a-2) = 5(a-2) - 1
.
Multiply out the right side so that the expression becomes:
.
f(a-2) = 5a - 10 -1
.
Finally, simplify the right side by combining the -10 and -1 to find that:
.
f(a-2) = 5a - 11
Hope this helps you to understand the basics of this problem.

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If f(x)= 4x -3, find the following:
f(-1)
What f(-1) tells you to do is to start with f(x) = 4x - 3, then replace every x you see
in it by (-1), and finally compute what the right side equals.
.
Let's change the problem a little and find f(-4). Go to the expression f(x) = 4x - 3 and
replace all the x's by -4. When you do the expression becomes:
.
f(-4) = 4(-4) - 3
.
When you multiply 4 times -4 you get -16. Substitute -16 for 4(-4) and the expression
becomes
.
f(-4) = -16 -3 and the two terms on the right combine so that the expression becomes:
.
f(-4) = -19
.
You can do the work to find f(-1). Follow the same process as the example.
.
I hope this helps you to understand what the problem really means.

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You have a good plan! The mid-point formula should get you the center of the circle. Then
use the distance formula for the length of the line that joins the two points. That will
give you the diameter of the circle so you need to divide it by 2 to get the radius.
.
The points are (-3,2) and (5,-6). The two values of x are -3 and + 5 so the distance
between them is 8 units. Divide that by 2 to get 4 units as the half-way point. Then either
add 4 units to -3 or subtract the 4 units from 5 to find that the mid-way point is at x = +1.
Next the two values of y are +2 and -6 so the distance between them is 8 units. Divide that
by 2 to get 4 units. Then either subtract the 4 units from +2 or add 4 units to -6 to
find the half-way point is at y = -2. Using that method you get the location of the midpoint
along the line joining the given points is at (+1, -2).
Since the x-distance between the points is 8 units and the y-distance between the points
is also 8 units, the distance between the two points (that is the hypotenuse) is:
.

.
The radius of the circle is half that or 5.65685 units.
.
You'd better work it yourself and check my answers. I did a lot in my head and it's late
at night. I'm prone to making mistakes without a coffee fix.

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I interpret your problem to be to solve for x in the equation:
.

.
and I assume you know the use of logarithms and exponents.
.
By using the power rule of exponents you can re-write the equation in the form:
.

.
so substitute 9 for to reduce the equation to:
.

.
Take the log of both sides to get:
.

.
But by the rules of exponents in logarithms, the exponent becomes a multiplier of the
log. x is the exponent, so it becomes the multiplier of log(9) to give:
.

.
To solve for x, divide both sides by log(9) to get:
.

.
Calculator time! log(12) = 1.079181246 and log(9) = 0.954242509. By dividing log(12) by
log(9) you find that x = 1.130929754.
.
That's the answer. By doubling it and raising 3 to that exponent on your calculator
you will find that the answer is, as it should be, 12.
.
And don't forget to round the answer to two decimal places as the problem requests.
.
Hope that I interpreted the problem correctly and that the approach I used was not beyond
where you are in your text.

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Without going into the geometric detail, if you draw a diagram of the two tangent circles
mark their centers, and draw a line connecting the two centers, the line will pass through
the point of tangency. Therefore, the line joining the two centers will be comprised
of the two radii. Call one of the radii x feet. Since the two radii will add together to be 10 feet,
the second radius will be 10-x feet.
.
It might not be a bad idea to make a sketch of the two circles with the line joining
their centers so you can understand what is happening.
.
You know that the Area of a circle is given by the equation:
.

.
where A represents the Area and R is the radius of the circle. So for this problem we
can compute the combined Area of the two circles by substituting x for one radius
and 10-x for the other and add the resulting two Areas. In equation form in which we
use "At" for total Area is:
.

.
Factor a out of each of the two terms on the right side:
.

.
Square the (10-x) term to make the equation:
.

.
Simplify the expression in the parentheses and rearrange it in descending powers of x:
.

.
At this point we use the fact that the combined Area of the two circles is: square feet.
by substituting this for At in the equation. Our equation then becomes:
.

.
Cancel the multipliers on both sides by dividing both sides by to get:
.

.
Subtract 52 from both sides and then transpose the equation:
.

.
Divide both sides by 2 to simplify the equation to:
.

.
This equation factors to:
.

.
And this equation will be true if either of the factors equals zero. Therefore, to solve
for x, set each factor equal to zero.
.
If then and if then . If then the
other radius is and if then the other radius is }.
From this, it is obvious that one radius must be 6 feet long and the other radius
must be 4 feet long.
.
Hope this helps to clarify the problem for you.

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.
Treat this as you would an equation and solve for Y by subtracting 6 from both sides.
The result is:
.

.
Graph this equation by making a graph of the equation Y=4. No matter what value X is, the
value of Y is always +4. This graph is a horizontal line that goes through +4 on the Y-axis
and runs infinitely far in both the positive and negative direction.
.
Notice that the equation tells you that the value of Y must at least equal
but can be above this line. You can shade in everything above the graphed line and just
including this line at the bottom of the shading, and the shaded area will be the region
where the value of Y can be.
.
Hope this helps you to understand the problem a little better.

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You multiply by just adding the two exponents to get .
.
Think about it. By the definition of exponents:
.
and

.
When you multiply the right sides of these two equations the problem becomes:
.

.
so you end up with 12 x's all multiplied together which is, by definition,
Hope this helps you understand why the exponent is 12.

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Here's one way you can think about these problems.
.
Let's begin by talking our way through:
.

Let's start by presuming that the > sign is an equal sign and the equation actually is
.

.
What does the graph of this equation look like? It is a horizontal line through the point
-2 on the y-axis. Does this make sense? What it says is that no matter what value you
select for x, the value of y will be -2.
.
The reality is that the equal sign was only put in to help us picture what is going on
with the graph. Now we can put the > back into the equation. Now we can tell that the
values of y must be greater than -2. This means that y is allowed to be any value
above the line that is the graph. You can shade that entire region, but only the region
that is ABOVE the line. y can be any value in the shaded region. However, y cannot have
the value -2 because y is only allowed to be GREATER than -2. Therefore, y can NOT be
on the line.
.
The next problem says that:
.

.
Like we did before, let's temporarily replace the > sign with an equal sign. This changes
the equation to:
.

.
This is in the slope-intercept form. Maybe you can picture the graph. It crosses the
y-axis at -2 and it slopes up and to the right at a rate of +2. That means for every 1 unit
you move horizontally to the right you go vertically up 2 units. You know that (0,-2) which
is the y-axis intercept is on the graph. You can easily find another point on the graph
by setting y = 0 in the line equation and then solving the equation for the corresponding
value of x:
.

.
When you solve this you find that x = 2 is the answer. Therefore, you know that (2,0) is a
second point on the graph. With the two points (0,-2) and (2,0) plotted you can draw a line
through them and you will have the graph of
.
At this point you should replace the = sign with the > sign to get back to the original
problem. This form tells you that y can only have values ABOVE the graph because y
must be greater than the values in the line. Shade the entire region above the line.
The shaded region is where values of y can be.
Finally, a little more complex (the last problem):
.

.
We can solve this for to make it easier for us to find the region where y is allowed to
exist just as we did before. We want to solve for +y. So let's multiply both sides of
the equation by -1. However, here's an important rule: whenever you multiply or divide both
sides of an inequality by a negative number, you must afterward reverse the direction of
the inequality.
.

.
Do the multiplication by -1 to get:
.

.
But don't forget that you have to reverse the direction of the inequality sign too. When
you do the inequality is now:
.

.
Now you can replace the inequality sign with an equal sign and solve the equation for y
just as you have always done. Begin by adding 4x to both sides to get:
.

.
Divide both sides by 6 to solve for y and get:
.

.
which becomes:
.

.
Graph this equation as you did previously. The slope is (2/3) and the y-axis intercept
is -2.
.
Now replace the = sign with the inequality sign pointing to the right so that the inequality
is now:
.

.
Again, shade the entire region ABOVE the graph of the right side of the equation.
That represents the place where y is allowed. y can NOT be on or below the graphed line.
.
After a little more practice you'll get familiar with this method and you can do things
faster and without thinking about it.
.
Hope this helps you with understanding the basic principles of doing problems such as
these inequalities.