# Algebra II : Solving Absolute Value Equations

## Example Questions

### Example Question #21 : Solving Absolute Value Equations

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 multiply both sides by 4.

The absolute value function turns whatever is inside positive, so this has two solutions. could be positive or negative 20, so we'll solve for both:

Next subtract 1 from both sides.

Then, divide by -3.

or

Now subtract 1 from both sides.

Finally, divide both sides by -3

### Example Question #22 : Solving Absolute Value Equations

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, divide both sides by 3.

The absolute value function makes whatever is inside positive, so this has two potential solutions. could either be positive or negative 9, so we have to solve for both:

or

### Example Question #23 : Solving Absolute Value Equations

If , find .

Explanation:

If , find .

Substitute negative seven in for every x.

Recall that the absolute value of a number always becomes positive.

### Example Question #24 : Solving Absolute Value Equations

Solve the absolute value equation.

and

and

and

Explanation:

Since absolute values are concerened with the distance from zero, you need two equations to show both possible solutions.

FIRST

SECOND

### Example Question #25 : Solving Absolute Value Equations

Solve:

or

or

Explanation:

To solve:

1. To clear the absolute value sign, we must split the equation to two possible solution. One solution for the possibility of a positive sign and another for the possibility of a negative sign:

or

2. solve each equation for :

or

### Example Question #26 : Solving Absolute Value Equations

Solve the following equation for b:

Explanation:

Solve the following equation for b:

Let's begin by subtracting 6 from both sides:

Next, we can get rid of the absolute value signs and make our two equations:

Next, add 13 to both sides

And divide by 5:

or

Next, add 13 to both sides

And divide by 5:

### Example Question #27 : Solving Absolute Value Equations

Solve for x:

no solution

no solution

Explanation:

Since the absolute value function only produces positive answers, an absolute value can never be equal to a negative number.

### Example Question #28 : Solving Absolute Value Equations

Solve the following equation:

Explanation:

To solve this you need to set up two different equations then solve for x.

The first one is:

where

The other equations is:

where

### Example Question #29 : Solving Absolute Value Equations

Find values of  which satisfy,

No solution

No solution

Explanation:

Recall the general definition of absolute value,

We will attempt two cases for the absolute value,

### Case 1:

Start by isolating the abolute value term,

Replace    with  and solve for :

### Case 2:

Replace   with   and solve for :

## Testing the Solutions

Whenever solving equations with an absolute value it is crucuial to check if the solutions work in the original equation. It often occurs that one or both of the solutions will not satisfy the original equation.

For instance, if we test  in the original equation we get,

This is clearly not true, since both sides are not equal, this rules out  as a solution. Similiarly, using   you can show that it also fails. Therefore, there is no solution.

### Example Question #30 : Solving Absolute Value Equations

What are all the possible values of  that fulfill the equation below?

and

and

only

and

and