Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities

$|x - 5| = x - 5$ If $x$ is less than 5, then $x - 5 < 0$, $|x - 5| = -(x - 5) = 5 - x$

Example: Find the solution set.

a. $\left|2x + 5\right| = 11$

b. $\left|3 - 2x\right| = -5$

c. $\left|2x - 3\right| = \left|x + 6\right|$

d. $2\left|x + 3\right| + 2 = 16$

Solution:

a. $\left|2x + 5\right| = 11$

$2x + 5 = 11$ or $2x + 5 = -11$

$2x = 6$ or $2x = -16$

$x = 3$ or $x = -8$

S.S. = {3, -8}

b. $\left|3 - 2x\right| = -5$

The absolute value principle reminds us that absolute value is always nonnegative. The equation $\left|3 - 2x\right| = -5$ has no solution. The solution set is $\phi$.

c. $\left|2x - 3\right| = \left|x + 6\right|$