To solve this problem, we want to set what's inside the absolute value signs equal to the positive and negative value on the right side of the equation. That's because the value inside the absolute value symbols could be equivalent to  or , and the equation would still hold true.

So let's set  equal to  and  separately and solve for our unknown.

First:

Second:

Therefore, our answers are  and . 

To solve, we replace each variable with the given value.

Simplify. Remember that terms inside of the absolute value are always positive.

Simplify the following:

Begin with basic subtraction:

Next, remember what we do with absolute value signs; we change negative values to positive values, and positive values remain positive.

So our answer is positive 51

When solving for x in the presence of absolute value, there are always two answers.

To eliminate the absolute value, the equation must be re-written two ways:

 and 

 and 

 and 

 and 

 and 

Substitute – 4 in for x. Remember that when a negative number is raised to the third power, it is negative. -  = – 64. – 64 – 36 = – 100. Since you are asked to take the absolute value of – 100 the final value of f(-4) = 100. The absolute value of any number is positive. 

Solve the first equation for x by dividing both sides of the equation by 6; the result is 7. Solve the second equation for k by dividing both sides of the equation by x, which we now know is 7. The result is 2/7.

Start by combining like terms.

4x + 5 = 13x + 4 – x – 9

4x + 5 = 12x – 5

–8x = –10

x = 5/4

First we solve for x.

Subtracting 3 from both sides gives us –3x < 17.

Dividing by –3 gives us x > –17/3.

–6 is less than –17/3.

The problem tells us that increasing x by twenty percent gives us the same thing that we would get if we decreased the product of four and x by five. We need to find expressions for these two situations, and then we can set them equal and solve for x.

Let's find an expression for increasing x by twenty percent. We could represent this as x + 20%x = x + 0.2x = 1.2x = 6x/5.

Let's find an expression for decreasing the product of four and x by five. First, we must find the product of four and x, which can be written as 4x. Then we must decrease this by five, so we must subtract five from 4x, which could be written as 4x - 5.

Now we must set the two expressions equal to one another.

6x/5 = 4x - 5

Subtract 6x/5 from both sides. We can rewrite 4x as 20x/5 so that it has a common denominator with 6x/5.

0 = 20x/5 - 6x/5 - 5 = 14x/5 - 5

0 = 14x/5 - 5

Now we can add five to both sides.

5 = 14x/5

Now we can multiply both sides by 5/14, which is the reciprocal of 14/5.

5(5/14) = (14x/5)(5/14) = x

25/14 = x

The answer is 25/14.