# Algebra II : Solving Absolute Value Equations

## Example Questions

### Example Question #31 : Solving Absolute Value Equations

Which of the following is in the set of all possible values for in the equation ?

The empty set.

Explanation:

First isolate the absolute value:

There are two possible values for : 7 and -7.

or

or

### Example Question #32 : Solving Absolute Value Equations

What is the set of all values for x that satisfy the equation ?

The empty set.

The empty set.

Explanation:

First isolate the absolute value.

There are no real numbers for which you can take the absolute value and get a negative number. Therefore, there are no values for x that will satisfy the equation.

### Example Question #33 : Solving Absolute Value Equations

Which of the following is the set of all values for  that satisfy the equation ?

The empty set.

Explanation:

First isolate the absolute value.

The two possible values for 2-3x are 5 and -5.

or

or

### Example Question #34 : Solving Absolute Value Equations

Solve for .

Explanation:

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract  on both sides.

Divide  on both sides.

### Example Question #35 : Solving Absolute Value Equations

Solve for .

Explanation:

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract  on both sides.

Distribute the negative sign to each term in the parenthesEs.

Divide both sides by .

### Example Question #36 : Solving Absolute Value Equations

Solve for .

Explanation:

When dealing with absolute value equations, we need to deal with negative values as well.

Subtract  on both sides.

Distribute the negative sign to each term in the parentheses.

Divide  on both sides.

### Example Question #37 : Solving Absolute Value Equations

Solve the following equation:

Explanation:

To solve absolute value equations, first we must isolate the absolute value:

Now that the absolute value is the only thing on the left hand side of the equation, we can solve for x.

Keep in mind that the absolute value makes whatever is inside of it a positive value. This means that and are both valid in solving the equation.

Setting both of these equal to 3 and solving for x, we get

As a check, plug each of these solutions back into the original equation and see if both solutions are valid!

### Example Question #38 : Solving Absolute Value Equations

Explanation:

To solve an absolute value equation, one must first rewrite the problem two different ways.

One is

and the other is

.

Then, solve each equation for v.

### Example Question #39 : Solving Absolute Value Equations

Solve for x:

Explanation:

To solve for x, we first must isolate the absolute value on one side of the equation:

Remember that an absolute value makes whatever is inside of it positive, so this equation is valid for both what is given inside the absolute value, and also the negative of what is inside:

Solve for x: