Remember that the absolute value of any number is its positive value, regardless of whether or not the number is negative before the absolute value is taken. We start by simplifying any expressions inside the absolute value signs:

Now we apply the absolute values and solve the expression:

In order to solve the given absolute value equation, we need to solve for  for the two ways in which this absolute value can be solved:

1.) 

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, there are two solutions to the absolute value equation:  and 

In order to solve the given absolute value equation, we need to solve for  in the two ways in which this absolute value can be solved:

1.)

2.) 

Solving Equation 1:

Solving Equation 2:

Therefore, there are two correct values of :  and .

In order to find the value of , we isolate the absolute value on one side of the equation:

At this point, however, we cannot solve the equation any further. By definition, absolute value can never equal a negative number; therefore, there is no value for  for this equation. 

There are two ways to solve the absolute value portion of this problem:

                or              

From here, you can solve each of these equations independently to arrive at the correct answer:

                        or                         

                           or                           

The solution is .

To solve an equation like |8x - 19| = 45, we set up two equations:

8x - 19 = 45 and 8x - 19 = -45.

Then it is simple arithmetic.

8x - 19 (+19) = 45 (+19)

8x/8 = 64/8

x = 8

8x - 19 (+19) = -45 (+19)

8x/8 = -26/8

x = -3.25

Therefore:

x = 8, -3.25

In order to solve for the values of , we need to isolate the variable:

When working with absolute value equations, however, we must remember that we are actually working with two equations:

  and 

Now we can solve for our values:

We can also write our answer as: 

Remember, when dividing by a negative number, switch the direction of the inequality sign.

can be rewritten as

, so

.

If , then

, or, equivalently, either 

 or .

Solve separately:

or 

, so the above two statements can be rewritten as 

 and 

 has no solution, since the absolute value of a number cannot be negative. 

 can be rewritten as

 and  

It is not necessary to solve these statements, as we can determine that the correct response is two solutions.

To solve absolute value equations, we must set up two equations: one where the solution is negative, and one where the solution is positive.

Assume Statement 1 alone.

  can be rewritten as

Therefore,  is positive.

Assume Statement 2 alone. The sign of  cannot be determined. For example, if , which is positive, then

.

If , which is not positive, then 

.