GMAT Math : Understanding absolute value

Example Questions

Example Question #11 : Absolute Value

The absolute value of negative seventeen is multiplied by a number that is three fewer than twelve. The resulting number is subtracted from negative six. What number is yielded at the end of this sequence of operations?

Explanation:

This is a problem where we need to use our translating skills. We are given a word problem and asked to solve it. To do so, we need to rewrite our word problem as an equation and then use arithmetic to find the answer. In these types of problems, the hardest step is usually translating correctly, so make sure to be meticulous and work step-by-step!

1)"The absolute value of negative seventeen": Recall that absolute value means that we will just change the sign to positive. Missing that will end up giving you the trap answer .

2)"is multiplied by a number which is three fewer than twelve." We need a number that is three fewer than twelve, so we need to subtract. Follow it up with multiplication and you get:

3)"The resulting number is subtracted from negative six." The key word here is "from"—make sure you aren't computing , which would result in another one of the trap answers!

Example Question #12 : Absolute Value

Solve .

or

or

or

Explanation:

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.)

2.)

Solving Equation 1:

Solving Equation 2:

Therefore, for  or .

Example Question #13 : Absolute Value

Solve .

or

or

or

Explanation:

Since we are solving an absolute value equation, , we must solve for both potential values of the equation:

1.)

2.)

Solving Equation 1:

Solving Equation 2:

Therefore, for  or .

Example Question #14 : Absolute Value

Define an operation  as follows:

For all real numbers ,

Evaluate:

Explanation:

Example Question #15 : Absolute Value

Define an operation  as follows:

For all real numbers ,

Evaluate

Explanation:

Example Question #16 : Absolute Value

Define an operation  as follows:

For all real numbers ,

Evaluate

Explanation:

Example Question #17 : Absolute Value

Define a function  to be

Give the range of the function.

Explanation:

An absolute value of a number must always assume a nonnegative value, so

, and

Therefore,

and the range of  is the set .

Example Question #18 : Absolute Value

Solve the following equation:

Explanation:

Before we apply the absolute value to the two terms in the equation, we simplify what's inside of them first:

Now we can apply the absolute value to each term. Remember that taking the absolute value of a quantity results in solely its value, regardless of what its sign was before the absolute value was taken. This means that that absolute value of a number is always positive:

Example Question #19 : Absolute Value

Give the range of the function

Explanation:

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both  and  are nonnegative, we can rewrite  as

, or

.

On  , this has as its graph a line with positive slope, so it is an increasing function. The range of this part of the function is , or, since

.

If , since  is negative and  is positive, we can rewrite  as

, or

is a constant function on this interval and its range is .

If , since both  and  are nonpositive, we can rewrite  as

, or

.

On  , this has as its graph a line with negative slope, so it is a decreasing function. The range of this part of the function is , or, since

.

The union of the ranges is the range of the function - .

Example Question #20 : Absolute Value

Give the range of the function

None of the other choices gives a correct answer.

None of the other choices gives a correct answer.

Explanation:

The key to answering this question is to note that this equation can be rewritten in piecewise fashion.

If , since both  and  are positive, we can rewrite  as

, or

,

a constant function with range .

If , since  is negative and  is positive, we can rewrite  as

, or

This is decreasing, as its graph is a line with negative slope. The range is ,

or, since

and

,

.

If , since both  and  are negative, we can rewrite  as

, or

,

a constant function with range .

The union of the ranges is the range of the function -  - which is not among the choices.

Tired of practice problems?

Try live online GMAT prep today.