# GMAT Math : Understanding absolute value

## Example Questions

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### Example Question #1 : Understanding Absolute Value

Solve .

or

or

or

or

Explanation:

really consists of two equations:

We must solve them both to find two possible solutions.

So  or  .

### Example Question #2 : Understanding Absolute Value

Solve .

Explanation:

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 : Understanding Absolute Value

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

Explanation:

Set  and solve for :

Rewrite as a compound equation and solve each part separately:

### Example Question #4 : Understanding Absolute Value

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

No such number exists.

Explanation:

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 : Understanding Absolute Value

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

Explanation:

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 : Understanding Absolute Value

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

No such number exists.

Explanation:

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 : Understanding Absolute Value

Solve for  in the absolute value equation

Explanation:

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 : Understanding Absolute Value

Solve the following equation:

Explanation:

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 : Understanding Absolute Value

Solve the absolute value equation for .

The equation has no solution

Explanation:

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 : Understanding Absolute Value

Which of the following could be a value of ?

Explanation:

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

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