# High School Math : Solving Absolute Value Equations

## Example Questions

### Example Question #1 : Solving Absolute Value Equations

Explanation:

Notice that the equation  has an  term both inside and outside the absolute value expression.

Since the absolute value expression will always produce a positive number and the right side of the equation is negative, a negative number must be added to the result of the absolute value expression to satisfy the equation. Therefore the term outside of the absolute value expression (in this case must be negative (meaning  must be negative).

Since  will be a negative number, the expression within the absolute value  will also be negative (before the absolute value is taken). It is thus possible to convert the original equation into an equation that treats the absolute value as a parenthetical expression that will be multiplied by , since any negative value becomes its opposite when taking the absolute value.

Simplifying and solving this equation for  gives the answer:

### Example Question #2 : Solving Absolute Value Equations

What are the possible values for ?

Explanation:

The absolute value measures the distance from zero to the given point.

In this case, since ,  or , as both values are twelve units away from zero.

Explanation:

### Example Question #4 : Solving Absolute Value Equations

Solve:

All real numbers

No solution

Explanation:

The absolute value can never be negative, so the equation is ONLY valid at zero.

The equation to solve becomes

### Example Question #12 : Algebra Ii

Solve for :

Explanation:

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

and

This gives us:

and

However, this question has an  outside of the absolute value expression, in this case . Thus, any negative value of  will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus,  is an extraneous solution, as  cannot equal a negative number.

Our final solution is then

Solve for .