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This is correct, except for a couple of equal signs that should have been plus signs. I fixed those for you. Now, since this is a fractional equation, you have to check the answers to make sure you didn't accidentally divide by zero. There are certain values of x that are NOT acceptable. In this problem, x cannot equal -4, so you have to reject this answer, since it makes a denominator zero--NOT allowed! The answer is NO SOLUTION!

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Did you want to factor this? Find two numbers whose product is 60 that will add up to 32! If you can't think of it, start with 1, and say
"1 times 60"
"2 times 30"
"3 times 20"
"4 times 15"
"5 times 12"
"6 times 10"
7 doesn't go, nor does 8, or 9, but
"10 times 6" repeats the "6 times 10", so that means that you have all the combinations.

Obviously the 2 times 30 is the combination you needed. Now

or
These answers are equally correct!

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You almost got it!! Just divide both sides of the equation by the coefficient of x:

so divide both sides by 7:

or

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Let x = the number.
= square of the number
= difference between a number and its square
= average of a number and it's square (add together and divide by 2)

The equation should look like this:


Multiply both sides of the equation by 2:



Set equal to zero, by subtracting x and from each side:



Factor out the common factor of x:

x= 0 or x= 3

There are two solutions.
x=0: The square is 0. The difference of the number and its square is zero, and the average of 0 and its square is also 0.

x= 3: The square is 9. The difference of the number and its square is 6, and the average of the number and its square is .

Both answers check.

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You must first find the slope of the given line by solving for y. Add 4x to each side:
-4x + 3y = 12
-4x + 4x + 3y = 4x + 12
3y = 4x + 12

Divide both sides by 3:



The slope of this line is , so the slope of ANY line PARALLEL to this line will also be . So the equation of any line is
y=mx+b

The equation of a line with slope is , where b is any number. It will also be the y-intercept of the line.

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Divide both sides by 2000:



In order to solve for t, where t is an exponent, you must take the ln of each side of the equation. It also works to take the log of each side--exactly the same answer!


This should bring back the laws of logarithms, in which the exponent becomes the coefficient:


Finally divide both sides by ln(1.06), and calculate with a calculator:
= 6.958515, which rounds off to about 6.96.

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What you have done so far looks right.
20x + 10 <= -1/2y

Now multiply both sides by -2, which of course reverses the inequality sign:
-2(20x + 10) >= -2(-1/2y)
-40x - 20 >= y

You probably don't want y on the right side so switch sides, but when you do this reverses the inequality again:
-40x - 20 >= y
y<=-40x-20

Is that what you wanted to do??

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Substitute the values of x to see what you get for f(x):
f(x) = -x - 3
f(-1) = -(-1) - 3 = -2
f(0) = -0 - 3= -3
f(4) = -4 - 3 =-7
f(7) = -7-3 = -10

Range is the set of all y-values, or in this case, all f(x) values, so
Range = { -2, -3, -7, -10}

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Draw the line y = 2, which is up two units on the y axis, and draw a HORIZONTAL LINE. Because you are graphing y > 2, you do NOT want to include the line, so draw a DOTTED LINE (not a solid line!). Next, since you are graphing y > 2, you will be shading ABOVE the line. If it had been y < 2, then you would have to shade BELOW the line.

I don't know how to draw this for you in the medium of algebra.com, so you'll have to draw it yourself. That is a DOTTED, HORIZONTAL LINE, UP 2 units, and shade everything above the line.

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GCF means the Greatest Common Factor, which means the largest factor that divides into all three terms. It may contain numbers and variables. First look at the numbers. The largest number that divides into 6, 12,and 18 is 6. There are NO variables that divide into all three terms, so the GCF = 6.

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Probability of a defective bulb =

The ratio of defective bulbs to the number of bulbs tested would be the fraction
== 3%

Out of 10280 bulbs, you would expect that approximately
or about 308 or 309 bulbs will be defective.

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If the given equation is y=3/2x -3, this means that the slope of the given line is . The slope of any line parallel to this line is also going to have a slope of , but that's another question. In this case, you need the slope of a line that is PERPENDICULAR to the given line. This slope would be the NEGATIVE RECIPROCAL of the slope which would be .

The equation of a line in slope intercept form is . So the slope of a line perpendicular to the given line must have slope of , so the equation would be , where b could be any value, since no other restrictions were given. Usually the problem requires that the line pass through a given point, in which case you must also solve for b. But this too is another question!

Final answer, , where b represents any number (which would also turn out to be the y intercept).

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Before beginning ANY cube root problem, be sure that you know your perfect cubes. That is,
, , ,

Now,

The idea is to somehow get the number in that radical in the denominator up to a perfect cube. When you think of the 3 inside the cube root sign, what perfect cube comes to mind associated with 3? The answer should be 27.

You need to get that denominator from 3 up to 27, by multiplying it by 9. In other words you must multiply the numerator and denominator by the cube root of 9.

Multiply numerator times numerator and denominator times denominator:
= = =

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The idea of solving equations is often a matter of "undoing" what has been done to the x, and usually in reverse order. Notice that in this equation, the x has a 4 added to it, the result is squared, and it equals 21. You need to UNDO those operations in reverse order. First undo the square, by square rooting both sides. Don't forget to put the PLUS or MINUS when you do this.




Next, undo the +4 by subtracting 4 from each side:



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Did you notice that this is a Quadratic Equation (see many of our lesson plans on solving quadratic equations!) Did you notice that this equation is already equal to zero? (This is VERY good!) Did you notice that there is a common factor of x? Then factor the common factor:
x(x-12)=0

Now, set each factor equal to zero.
x= 0 and x-12= 0
x=0 and x= 12

Wasn't that easy??

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Is this a complex fraction that looks like this:


If so, then notice that you have a numerator which is a fraction and a denominator which is a fraction. Notice also that the middle fraction line separating the numerator and the denominator is the longest line. Now, just "unstack" the problem by writing this in the form of a
(NUMERATOR) divided by (DENOMINATOR).

Now, invert the second fraction and multiply. While you are at it, factor all the trinomials:


Divide out the (x-7) and the (x-2) factors, and you should be left with for your final answer.

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Get the y terms on one side (the left side) and all the other terms to the right side, but adding -2x to each side:
2x + 3y = 6
2x - 2x + 3y = -2x + 6
3y = -2x + 6
Next, divide both sides by 3 to undo the 3. It can be done in two ways. Here is the first way:



OR the second way:



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Please see my recent Lesson Plan on Graphing and Slope of the Line for more details and examples. In this equation, the y intercept is 1, and the slope is -4 or -4/1, which means a rise of -4 and a run of 1.

To graph this start by graphing the y-intercept by going 1 unit up on the y-axis, and put the first point here. Next, with your pencil on this point on the y-axis, move DOWN 4, then go over 1 unit to the right, and put the next point. Then draw a line connecting the two points. It should look like this:


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Multiply both sides of the equation by the denominator, which is 3, to eliminate the denominator in the equation:



Now to solve for h, you need to "undo" everything that has been done to the h. That is, you multiplied by all those other factors: . So you need to DIVIDE by those factors in order to undo the multiplications:



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Begin with 24 L of .20 antifreeze
Drain x L of .20 antifreeze
Add x L of 1.00 antifreeze
Equals 24 L. of .40 antifreeze

Equation:
.20(24) - .20x + 1.00x = .40(24)
4.80 + .80x = 9.60
4.80-4.80 +.80x = 9.60 - 4.80
.80x = 4.80
liters

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Would you like an equation, or shall we do it the easy way?

The easy way is to notice that 4 dozen cookies require 2 cups, so 36 cookies would be 3 dozen. That's 3/4 as many cookies, which will require 3/4 as many chocolate chips.

So, just take 3/4 of 2 cups, which would be or a cup and a half of chocolate chips.

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2 1/2 dozen muffins would be 24 + 6 or 30 muffins with 600 g of flour.
Let x = number of muffins made with 900 g of flour.

Set up a ratio with muffins divided by flour:


Remember that means that

In the same way, cross-multiply
to obtain:



Divide by 600:
muffins.

HOWEVER, the EASY way to do this, in this particular case, is to notice that you are comparing 600 g of flour with 900 g of flour. This is half again as much as 600 g., so you can make half again as many muffins. Half of 30 is 15, so add 30+15= 45 muffins. Much easier, right???

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Let x = time of the second plane flying at 500 mph.
x+1 = time of the first plane flying at 375 mph.

Distance of first plane = Distance of second plane
R*T = R*T
500x = 375(x+ 1)
500x = 375x + 375
500x - 375x = 375x - 375x + 375
125x = 375

hours = time for second plane to catch the first plane.

Check:
Distance of first plane = 4 *375= 1500 miles
Distance of second plane = 3*500 = 1500 miles

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The first step is to rationalize the first denominator by multiplying numerator and denominator of the first fraction by :





Notice the common denominator!!


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I have a feeling that whoever made up or assigned this problem had in mind to use the distibutive property to factor out the common factor of 13.

7*13+8*13+5*13 = 13*(7+8+5), which is 13*20 or 260, and you can almost do it in your head.

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This one is easier than it looks. First look at the second equation. You can see by looking at this equation that .

Knowing that t=-4, solve for s in the third equation:
6s+5t= 10
6s+5(-4) = 10
6s-20=10
6s=30
s=5

Knowing that s=5, solve for r in the first equation:
5r + 2s=0
5r+2(5)=0
5r +10=0
5r= -10
r=-2

(r,s,t) = (-2, 5. -4)

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Assuming that you mean , the first step is to set the equation equal to zero to write , and graph . The solution will be the value(s) of x where this graph crosses the x axis. You can assume because of the square root that . The value of x where the graph crosses the x axis is the solution for this equation:


The normal graph for x from -10 to 10 may NOT work here. You may need to use a value of x larger than 10. Let's try graphing up to x= 20 and see if the graph crosses the x axis. It turns out that x= 11 is the value that works, so this is the solution. X=11, and it checks also!

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Do you mean ?? If so, then the first step in adding fractions is to factor the denominators so as to find the LCD.

and the LCD = (a-2)(a+2).

In order to get the LCD in the first fraction, you already have the (a+2) factor, but you will need to multiply numerator and denominator of this first fraction by (a-2). The second fraction already has the LCD, so leave it alone.



Now you have an LCD of (a-2)(a+2), and this becomes THE denominator of the fraction:


Now, just add numerators together as follows:


Simplify:

The numerator DOES factor, so you must factor the numerator in order to reduce the fraction.


Divide out the (a+2) factor in the denominator with one of the numerator factors:
FINAL ANSWER!!

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I'm assuming that your problem is .

Since this is a binomial denominator (two terms!), you must multiply numerator and denominator by the conjugate of the denominator--that is, the same as the denominator, but with the oppposite sign in the middle.

It looks like this:
*

When you expand the denominator, the middle terms subtract out (as they ALWAYS do!), and the denominator is just . This has no radicals, so it is said to be rationalized.

Final answer:

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Solve for y:
3x+4y = 12

Subtract 3x from each side:
3x-3x + 4y = -3x + 12
4y = -3x + 12

Divide both sides by 4 (it's best in this case to divide each term by 4):



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Factoring and expanding are exactly the opposite operation. To factor means to express as a product. To expand means to multiply something out.

For examples, to factor 15 means to write it as a product 3*5.
To factor 5x+ 20 means to write 5(x+4), which is written as a product.
To factor means to write it as a product .
To expand 5(x+4) means to remove the parentheses and multiply it out: 5x + 20.
To expand means to multiply it out: .

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