# SAT II Math I : Absolute Value

## Example Questions

### Example Question #68 : Sat Subject Test In Math I

Define an operation  on the set of real numbers as follows:

For any two real numbers

Evaluate the expression

Explanation:

Substitute  in the expression:

### Example Question #69 : Sat Subject Test In Math I

Simplify the following expression:

Explanation:

To simplify, we must first simplify the absolute values.

Now, combine like terms:

### Example Question #70 : Sat Subject Test In Math I

Solve for .

Explanation:

To solve for x we need to make two separate equations. Since it has absolute value bars around it we know that the inside can equal either 7 or -7 before the asolute value is applied.

### Example Question #1 : Absolute Value

The absolute value of a negative can be positive or negative. True or false?

False

True

False

Explanation:

The absolute value of a number is the points away from zero on a number line.

Since this is a countable value, you cannot count a negative number.

This makes all absolute values positive and also make the statement above false.

### Example Question #2 : Absolute Value

Which of the following absolute value equations has the same solution set?

None of the other choices gives the correct response.

Explanation:

Rewrite the quadratic equation in standard form by subtracting  from both sides:

Factor this as

where the squares represent two integers with sum  and product 14. Through some trial and error, we find that  and  work:

By the Zero Product Principle, one of these factors must be equal to 0.

If  then ;

if  then .

The given equation has solution set , so we are looking for an absolute value equation with this set as well.

This equation can take the form

This can be rewritten as the compound equation

Adding  to both sides of each equation, the solution set is

and

Setting these numbers equal in value to the desired solutions, we get the linear system

Backsolving to find :

The desired absolute value equation is .

### Example Question #3 : Absolute Value

What is the value of: ?

Explanation:

Step 1: Evaluate ...

Step 3: Define absolute value...

The absolute value of any value  is always positive, unless there is an extra negation outside (sometimes)..

Step 4: Evaluate...

### Example Question #4 : Absolute Value

Solve:

Explanation:

Divide both sides by negative three.

Since the lone absolute value is not equal to a negative, we can continue with the problem.  Split the equation into its positive and negative components.

Evaluate the first equation by subtracting one on both sides, and then dividing by two on both sides.

Evaluate the second equation by dividing a negative one on both sides.

Subtract one on both sides.

Divide by 2 on both sides.
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