### All GMAT Math Resources

## Example Questions

### Example Question #1 : Absolute Value

Solve .

**Possible Answers:**

or

or

or

**Correct answer:**

or

really consists of two equations:

We must solve them both to find two possible solutions.

So or .

### Example Question #2 : Absolute Value

Solve .

**Possible Answers:**

**Correct answer:**

It's actually easier to solve for the complement first. Let's solve . That gives -3 < 2x - 5 < 3. Add 5 to get 2 < 2x < 8, and divide by 2 to get 1 < x < 4. To find the real solution then, we take the opposites of the two inequality signs. Then our answer becomes .

### Example Question #3 : Absolute Value

Give the -intercept(s), if any, of the graph of the function in terms of .

**Possible Answers:**

**Correct answer:**

Set and solve for :

Rewrite as a compound equation and solve each part separately:

### Example Question #4 : Absolute Value

A number is ten less than its own absolute value. What is this number?

**Possible Answers:**

No such number exists.

**Correct answer:**

We can rewrite this as an equation, where is the number in question:

A nonnegative number is equal to its own absolute value, so if this number exists, it must be negative.

In thsi case, , and we can rewrite that equation as

This is the only number that fits the criterion.

### Example Question #5 : Absolute Value

If , which of the following has the greatest absolute value?

**Possible Answers:**

**Correct answer:**

Since , we know the following:

;

;

;

;

.

Also, we need to compare absolute values, so the greatest one must be either or .

We also know that when .

Thus, we know for sure that .

### Example Question #6 : Absolute Value

Give all numbers that are twenty less than twice their own absolute value.

**Possible Answers:**

No such number exists.

**Correct answer:**

We can rewrite this as an equation, where is the number in question:

If is nonnegative, then , and we can rewrite this as

Solve:

If is negative, then , and we can rewrite this as

The numbers have the given characteristics.

### Example Question #7 : Absolute Value

Solve for in the absolute value equation

**Possible Answers:**

None of the other answers

**Correct answer:**

None of the other answers

The correct answer is that there is no .

We start by adding to both sides giving

Then multiply both sides by .

Then divide both sides by

Now it is impossible to go any further. The absolute value of any quantity is always positive (or sometimes ). Here we have the absolute value of something equaling a negative number. That's never possible, hence there is no that makes this a true equation.

### Example Question #8 : Absolute Value

Solve the following equation:

**Possible Answers:**

**Correct answer:**

We start by isolating the expression with the absolute value:

becomes

So: or

We then solve the two equations above, which gives us 42 and 4 respectively.

So the solution is

### Example Question #9 : Absolute Value

Solve the absolute value equation for .

**Possible Answers:**

The equation has no solution

None of the other answers.

**Correct answer:**

We proceed as follows

(Start)

(Subtract 3 from both sides)

or (Quantity inside the absolute value can be positive or negative)

or (add five to both sides)

or

Another way to say this is

### Example Question #10 : Absolute Value

Which of the following could be a value of ?

**Possible Answers:**

**Correct answer:**

To solve an inequality we need to remember what the absolute value sign says about our expression. In this case it says that

can be written as

Of .

Rewriting this in one inequality we get:

From here we add one half to both sides .

Finally, we divide by two to isolate and solve for m.

Only is between -1.75 and 2.25