Chapter 1.4, Problem 57E

To determine

To explain:the reason of having no solution, one solution, or two solutions

Answer to Problem 57E

If the absolute value is compared to 0, there will be one solution.

If the absolute value is greaterthan zero come up there will be one or two solutions.

The equation has no solution ifit is equal to a negative number.

Explanation of Solution

Given:

Absolute value equations can have no solution, one solution, or two solutions.

Calculation:

Example:

The absolute value of anexpression must be greaterthan or equal to 0

or

The equation has no solution ifit is equal to a negative number.

  $|x+3|=-3$

  $|x+3|+6=3$

Isolate the absolute value

  $|x+3|=-3$

Example:

If the absolute value is compared to 0, there will be one solution.

  $\begin{array}{c}|x+3|=0\\ x+3=0\\ x=-3\end{array}$

Example:

If the absolute value is greaterthan zero come up there will be one or two solutions.

  $|x+3|=6$

  $x+3=6$

  $x=3$

And

  $x+3=-6$

  $x=-9$

This equation has two solutions.

Another example:

  $\begin{array}{l}|x-3|=|x+2|\\ x-3=x+2\\ -3\ne 2\end{array}$

And

  $x-3=-(x+2)$

  $\begin{array}{l}x-3=-x-2\\ 2x-3=-2\\ 2x=1\end{array}$

  $x=\frac{1}{2}$

This equation has one solution

Conclusion:

The reason of having no solution, one solution, or two solutions is explained.