# GMAT Math : Understanding absolute value

## Example Questions

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### Example Question #31 : Understanding Absolute Value

If , then how many possible values of  are there?

Zero

Four

Two

Three

One

Two

Explanation:

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.

### Example Question #32 : Absolute Value

Solve for

Explanation:

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

### Example Question #33 : Absolute Value

True or false:  is a positive number.

Statement 1:

Statement 2:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

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

.

### Example Question #34 : Absolute Value

How many values of  make

a true statement?

Four

Two

Three

One

None

Two

Explanation:

, so we want the number of values of  for which

.

, so

Therefore, if , then

Either

, in which case , or

, in which case .

The correct choice is therefore two.

### Example Question #35 : Absolute Value

How many values of  make

a true statement?

Two

None

One

Three

Four

None

Explanation:

, so we want the number of values of for which

, so either

or

If the first equation is true, then

and

.

If the second equation is true, then

and

.

In each situation, the absolute value of an expression would be negative; since the absolute value of an expression cannot be negative, no solution is yielded.

There are no values of that make true; the correct response is zero.

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