# Algebra II : Solving Absolute Value Equations

## Example Questions

### Example Question #41 : Solving Absolute Value Equations

Solve:

Explanation:

Break up the absolute value into its positive and negative components.

The equations are:

Solve the first equation.  Add 4 on both sides.

Divide by three on both sides.

The first solution is:

Solve the second equation.  Divide by negative one on both sides to move the negative from left to right.

Divide by three on both sides.

The solutions are:

### Example Question #42 : Solving Absolute Value Equations

Solve:

Explanation:

To solve, we must first isolate the absolute value on one side:

Next, we must remember that the equation holds true for both the positive and negative of what is inside the absolute value sign, because the absolute value makes whatever is inside of it positive:

Solving for x, we get

One can plug the solutions back into the original equation to confirm.

### Example Question #43 : Solving Absolute Value Equations

Solve the equation:

Explanation:

Solve by first dividing negative two on both sides.  This will isolate the absolute value term.

Do NOT continue with this problem!  There are no cases where an absolute value of a number will equal to negative four.

The answer to this equation is:

### Example Question #44 : Solving Absolute Value Equations

Solve:

Explanation:

Break up the absolute value and split this equation into its positive and negative components.

Solve the first equation.  Add 18 on both sides.

Simplify both sides.  One of the answers is:

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

The equation becomes:

The other solution is:

### Example Question #45 : Solving Absolute Value Equations

Solve the absolute value:

Explanation:

Break up this absolute value into its positive and negative components.

Evaluate the first equation.  Subtract one from both sides and simplify.

Divide by eight on both sides.

Simplify both sides.  There is no need to switch the sign because we are not dividing both sides by a negative number.

The first solution is:

Evaluate the second equation by dividing both sides by a negative one.

Simplify and switch the direction of the sign.

Subtract one from both sides.

Divide by eight on both sides.

The second solution is:

### Example Question #46 : Solving Absolute Value Equations

Solve the inequality:

Explanation:

According to absolute value rules, an absolute value will be converted into a positive number.

In the given problem  ,

we are asked to determine a certain  value inside an absolute value that must be less than negative twenty five.  However, there are no values of an absolute value that will give a negative number.

### Example Question #47 : Solving Absolute Value Equations

Solve the absolute value equation:

Explanation:

Break up the absolute value and rewrite the equations in their positive and negative components.

The equations are:

Evaluate the first equation.  Add 10 on both sides.

Divide by 8 on both sides.

The first solution is:

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

Divide by 8 on both sides.

The second solution is:

### Example Question #48 : Solving Absolute Value Equations

Solve the absolute value equation:

Explanation:

Recall that the absolute value sign will convert any value to a positive sign.  There will be no occurrences of  that will evaluate into a negative one as a final solution.

There are no solutions for this equation.

### Example Question #49 : Solving Absolute Value Equations

Solve the equation:

Explanation:

Break up the absolute value into its positive and negative components.

The equations become:

Solve the first equation.  Subtract three on both sides.

Divide by nine on both sides.

The first solution is .

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

Subtract three on both sides.

Divide by nine on both sides.

Reduce the fractions.

The second solution is:

### Example Question #41 : Solving Absolute Value Equations

Solve the following equation in real numbers:

Explanation:

Recall the definition of the absolute value operation:

whenever , and  whenever .

Hence, this equation can be split into two separate equations:

and .

Solving the first equation yields:

,

Solving the second equation yields:

,

Substituting each of these solutions into the original equation yields a true statement. However, we are only looking for real values of , and so must disregard the complex solutions . Hence, the desired solutions to the equation are

.