### All GMAT Math Resources

## Example Questions

### Example Question #11 : Absolute Value

Define a function to be

Give the range of the function.

**Possible Answers:**

**Correct answer:**

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

, and

Therefore,

and the range of is the set .

### Example Question #12 : Absolute Value

Solve the following equation:

**Possible Answers:**

**Correct answer:**

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 #451 : Algebra

Give the range of the function

**Possible Answers:**

**Correct answer:**

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 #13 : Understanding Absolute Value

Give the range of the function

**Possible Answers:**

None of the other choices gives a correct answer.

**Correct answer:**

None of the other choices gives a correct answer.

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.

### Example Question #451 : Algebra

Simplify the following expression:

**Possible Answers:**

**Correct answer:**

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 #451 : Algebra

Solve the following inequality:

**Possible Answers:**

**Correct 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 :

### Example Question #451 : Algebra

Solve the following inequality:

**Possible Answers:**

or

or

or

**Correct answer:**

or

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 #452 : Algebra

**Possible Answers:**

**Correct answer:**

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 #21 : Absolute Value

Solve for :

**Possible Answers:**

Not enough information to solve

and

and

**Correct answer:**

and

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

and

Not enough information to solve

**Correct answer:**

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 .

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