# GMAT Math : Understanding absolute value

## Example Questions

### Example Question #21 : Absolute Value

Simplify the following expression:

Possible Answers:

Correct answer:

Explanation:

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 .

### Example Question #22 : Absolute Value

Solve the following inequality:

Possible Answers:

Correct answer:

Explanation:

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 :

### Example Question #23 : Absolute Value

Solve the following inequality:

Possible Answers:

or

or

or

Correct answer:

or

Explanation:

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

### Example Question #24 : Absolute Value

Possible Answers:

Correct answer:

Explanation:

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:

### Example Question #25 : Absolute Value

Solve for

Possible Answers:

and

and

Not enough information to solve

Correct answer:

and

Explanation:

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

### Example Question #26 : Absolute Value

Solve for

Possible Answers:

and

Not enough information to solve

and

Correct answer:

and

Explanation:

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 .

### Example Question #27 : Absolute Value

Solve the following absolute value equation for

Possible Answers:

No value for

or

or

Correct answer:

No value for

Explanation:

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.

### Example Question #28 : Absolute Value

Find the possible values of :

Possible Answers:

Correct answer:

Explanation:

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 .

### Example Question #29 : Absolute Value

Solve for :

Possible Answers:

Correct answer:

Explanation:

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

### Example Question #30 : Absolute Value

Solve.

Possible Answers:

Correct answer:

Explanation:

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.

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