# GMAT Math : Algebra

## Example Questions

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### Example Question #461 : Algebra

Solve the following absolute value equation for

or

or

No value for

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 #23 : Understanding Absolute Value

Find the possible values of :

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 #462 : Algebra

Solve for :

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 #1541 : Problem Solving Questions

Solve.

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.

### Example Question #1541 : Gmat Quantitative Reasoning

If , then how many possible values of  are there?

Two

Four

One

Zero

Three

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 #31 : 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 #463 : Algebra

True or false:  is a positive number.

Statement 1:

Statement 2:

EITHER STATEMENT ALONE provides 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.

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.

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 #1551 : Problem Solving Questions

How many values of  make

a true statement?

One

Four

None

Two

Three

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 #461 : Algebra

How many values of  make

a true statement?

Four

None

Three

One

Two

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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