# SAT II Math II : Absolute Value

## Example Questions

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Define .

Evaluate .

Explanation:

### Example Question #1 : Absolute Value

Define .

Order from least to greatest:

Explanation:

, or, equivalently,

From least to greatest, the values are

### Example Question #31 : Mathematical Relationships

Define an operation  as follows:

For all real numbers ,

Evaluate .

Undefined.

Explanation:

### Example Question #1 : Absolute Value

Define an operation  as follows:

For all real numbers ,

If , which is a possible value of ?

Explanation:

, so

can be rewritten as

Therefore, either  or . The correct choice is .

### Example Question #5 : Absolute Value

Define .

How many values are in the solution set of the equation  ?

Three solutions

No solutions

One solution

Infinitely many solutions

Two solutions

No solutions

Explanation:

We can rewrite this function as a piecewise-defined function by examining three different intervals of -values.

If , then

and ,

and this part of the function can be written as

If  under this definition, then

However, , so this is a contradiction.

If , then

and ,

and this part of the function can be written as

This yields no solutions.

If , then

and ,

and this part of the function can be written as

If  under this definition, then

However, , so this is a contradiction.

has no solution.

### Example Question #6 : Absolute Value

Define .

How many values are in the solution set of the equation  ?

Two solutions

One solution

No solutions

Infinitely many solutions

Three solutions

Infinitely many solutions

Explanation:

We can rewrite this function as a piecewise-defined function by examining three different intervals of -values.

If , then

and ,

and this part of the function can be written as

If , then

and ,

and this part of the function can be written as

If , then

and ,

and this part of the function can be written as

The function can be rewritten as

As can be seen from the rewritten definition, every value of  in the interval  is a solution of , so the correct response is infinitely many solutions.

### Example Question #7 : 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:

Solve this equation using the  method. We are looking for two integers whose sum is  and whose product is ; by trial and error we find they are . The equation becomes

Solving using grouping:

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

Either

Or

The given quadratic 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 #2 : Absolute Value

Solve .

Explanation:

First we need to isolate the absolute value term.  We do with using some simple algebra:

Now we solve two equations, one with the right side of the equation positive, one negative.  Let's start with the positive:

And now the negative:

### Example Question #2 : Absolute Value

Solve .

No solutions

No solutions

Explanation:

First, we have to isolate the absolute value:

Let's take a look at our equation right now.  It's saying that the absolute value has to be a negative number, which isn't possible.  So no solutions exist.

Solve .