# Algebra II : Solving Absolute Value Equations

## Example Questions

### Example Question #2 : Absolute Value Inequalities

Solve the inequality.

Explanation:

Remove the absolute value by setting the term equal to either or . Remember to flip the inequality for the negative term!

Solve each scenario independently by subtracting from both sides.

### Example Question #12 : Solving Absolute Value Equations

Solve for  :

Explanation:

The absolute value of any number is nonnegative, so  must always be greater than . Therefore, any value of  makes this a true statement.

### Example Question #11 : Solving Absolute Value Equations

Solve for .

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

. S

ubtract  on both sides of all equations.

### Example Question #12 : Solving Absolute Value Equations

Solve for .

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

Add  to both sides of both equations.

### Example Question #13 : Solving Absolute Value Equations

Solve for .

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

Multiply both sides by  for both equations.

### Example Question #14 : Solving Absolute Value Equations

Solve for .

Explanation:

When solving for absolute values, remember there's an equation for positive value and another equation for negative value.

Divide both sides by  for both equations.

### Example Question #13 : Solving Absolute Value Equations

Solve:

Explanation:

To solve absolute value equations, you have to set up the equation two different ways to account for two answers.

The first is how you see it:

.

The second is by making the answer negative:

.

Next solve each one for x. The first equation yields x to be 7. The second one gives you x=-6. Those are your two answers.

### Example Question #16 : Solving Absolute Value Equations

Solve for :

No solution

No solution

Explanation:

Given

first subtract 8 from both sides to get  on the right.

Based upon what we know about absolute value functions, it is impossible for an absolute value to be negative. So it is also impossible for an absolute value to be less than a negative number, which in this case is

The only answer can be No Solution.

### Example Question #17 : Solving Absolute Value Equations

Solve for :

Explanation:

Solve using inverse opperations:

Divide both sides by 2.

Because the absolute value function always makes whatever is inside positive, this gives us two solutions. could be either or , so we'll solve for both:

or

### Example Question #11 : Absolute Value

Solve for :

Explanation:

In order to solve the equation, we have to isolate the variable. We do this by performing the same operation to either side of the equation.

First add 4 to both sides

The absolute value function makes whatever is inside positive, which means that there are two possible solutions. could either be or
so we'll solve for both

or