This question plays a few tricks dealing with absolute values. To begin, we can recognize that any negative sign within an absolute value can basically be rendered positive. So this:

becomes:

In this case, we still have a negative that was outside of the absolute value sign. This term will stay negative, so we get:

This makes our answer .

To solve this absolute value inequality, we must remember that the absolute value of a function that is less than a certain number must be greater than the negative of that number. Using this knowledge, we write the inequality as follows, and then perform some algebra to solve for :

To solve this absolute value inequality, we must remember that the absolute value of a function that is greater than a certain number is also less than the negative of that number. With this in mind, we rewrite the inequality as follows and then solve for the possible intervals of :

   or   

   or   

   or   

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.