# Algebra II : Solving Absolute Value Equations

## Example Questions

### Example Question #61 : Solving Absolute Value Equations

Solve the absolute value:

Explanation:

In order to solve this equation, we will need to eliminate the absolute value signs and rewrite the left term in its positive and negative components.

Solve the first equation.  Add eight on both sides.

Divide by three on both sides.

The first solution is:

Divide the second equation by negative one.  Dividing a negative value will require switching the direction of the sign.

Divide by three on both sides.

The second solution is:

### Example Question #62 : Solving Absolute Value Equations

Explanation:

Recall that when dealing with absolute value, you have to write the inequality two different ways to solve it:

1)

2)

Now, solve each one

### Example Question #63 : Solving Absolute Value Equations

Solve:

Explanation:

Start by isolating the absolute value.

Subtract seven on both sides.

Divide by three on both sides.

There is no value of  in an absolute value that will give a negative value since the absolute value will convert all numbers to positive.

### Example Question #64 : Solving Absolute Value Equations

Solve for x:

Explanation:

What is inside the absolute value brackets could be positive or negative.

Positive:

Negative:

### Example Question #65 : Solving Absolute Value Equations

Solve:

Explanation:

Separate the absolute value and solve both the positive and negative components of the absolute value.

Solve the first equation.  Add  on both sides.

Divide by five on both sides.

One of the solutions is  after substitution is valid.

Subtract  on both sides.

If we substitute this value back to the original equation, the equation becomes invalid.

### Example Question #66 : Solving Absolute Value Equations

Solve the absolute value equation:

Explanation:

Divide by two on both sides to isolate the absolute value.

The equation becomes:

Split this equation into its positive and negative components.

Solve the first equation.  Add three on both sides.

Divide by three on both sides.

The first solution is:

Evaluate the second equation.  Divide by negative one on both sides.

Divide by three on both sides.

The second solution is:

### Example Question #61 : Absolute Value

Evaluate:

Explanation:

Isolate the absolute value by dividing both sides by negative three.

The equation becomes:

Recall that an absolute value cannot be negative since the absolute value will change negative values to positive.

### Example Question #68 : Solving Absolute Value Equations

Solve:

Explanation:

Subtract  on both sides.

Evaluate the positive and negative components of the absolute value.  The two equations are:

Evaluate the first equation.  Add  on both sides.

There is no solution for the first equation.

Evaluate the second equation.  Simplify the left side by distributing negative one through the binomial.

The equation becomes:

Divide by six on both sides and reduce the fraction.  Verify that the answer will satisfy the original equation.

### Example Question #69 : Solving Absolute Value Equations

Solve the absolute value equation:

Explanation:

Evaluate by adding six on both sides.

Divide by negative two on both sides.

The equation becomes:

Recall that absolute values will convert all negative values to positive.  No matter what  would evaluate to be, it will never become negative half.

### Example Question #70 : Solving Absolute Value Equations

Solve:

Explanation:

Divide both sides by eight.

Split the absolute value into its positive and negative components.

Solve the first equation.  Multiply both sides by eight to eliminate the fraction.

Use distribution to simplify.

Add 24 on both sides, and then divide both sides by eight.

The first solution is .

Solve the second equation.  Divide by negative one on both sides.

The equation becomes:

Add three on both sides.  This is the same as adding  on both sides.