Glossary Checklist FAQ

B. Solving Absolute Value Equations Algebraically

 Investigation


Solving Absolute Value Equations Algebraically


To solve equations, you have typically been able to use inverse operations to isolate a variable. That is, you add to “undo” subtraction and divide to “undo” multiplication. Absolute values can be defined piecewise as \(\left| x \right| = \left\{\begin{align}
&x{\rm \thinspace {for}} \thinspace x \ge 0 \\
-&x{\rm \thinspace {for}} \thinspace x < 0 \end{align} \right.\), so the operation to “undo” an absolute value is not constant.

  1. Consider the equation \(\left|x\right| = 7\).

    1. Suppose \(x\) is positive.

      1. What can be done to both sides of the equation to “undo” the absolute value sign?

      2. What value must \(x\) be?

    2. Suppose \(x\) is negative.

      1. What can be done to both sides of the equation to “undo” the absolute value sign?

      2. What value must \(x\) be?

    3. What are the solutions to \(\left|x\right| = 7\)?

  2. Use the reasoning from question 1 to solve \(\left|2x + 5\right| = 17\).

  3. Use the reasoning from question 1 to solve \(\left|x^2 +2x - 4\right| = 4\).