### All SAT II Math II Resources

## Example Questions

### Example Question #1 : Absolute Value

Define .

Evaluate .

**Possible Answers:**

**Correct answer:**

### Example Question #2 : Absolute Value

Define .

Order from least to greatest:

**Possible Answers:**

**Correct answer:**

, or, equivalently,

From least to greatest, the values are

### Example Question #3 : Absolute Value

Define an operation as follows:

For all real numbers ,

Evaluate .

**Possible Answers:**

Undefined.

**Correct answer:**

### Example Question #4 : Absolute Value

Define an operation as follows:

For all real numbers ,

If , which is a possible value of ?

**Possible Answers:**

**Correct answer:**

, 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 ?

**Possible Answers:**

Three solutions

Infinitely many solutions

One solution

Two solutions

No solutions

**Correct answer:**

No solutions

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 ?

**Possible Answers:**

Two solutions

One solution

Three solutions

No solutions

Infinitely many solutions

**Correct answer:**

Infinitely many solutions

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

Consider the quadratic equation

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

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

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

Adding and solving for :

Backsolving to find :

The desired absolute value equation is .

### Example Question #8 : Absolute Value

Solve .

**Possible Answers:**

**Correct answer:**

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:

So our answers are:

### Example Question #9 : Absolute Value

Solve .

**Possible Answers:**

No solutions

**Correct answer:**

No solutions

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.

### Example Question #10 : Absolute Value

Solve .

**Possible Answers:**

No solutions

**Correct answer:**

First, we need to isolate the absolute value:

Because the equation is set equal to , we can drop the absolute value symbols and solve normally:

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