# Section 1.3 Absolute Value Equations and Inequalities

Section Objectives

1. Solve absolute value equations.
2. Solve absolute value inequalities.

### Absolute Value

The absolute value of a number, $x$, is that number's distance from zero on the number line. The absolute value of $x$ is written $|x|$, and this quantity is always either positive or zero. In fact,

• $|x| = x$ if $x$ itself is positive or zero, and
• $|x| = -x$ (think the opposite of $x$) if $x$ is negative.

Based on the definition of absolute value, it is clear that $|x| = x$ or $-x$, depending on whichever is positive. If we're not sure which is positive, we must consider both options.

For example...

If $|x|=5$, then $x$ must either be $5$ or $-5$.

### Absolute Value Equations

Suppose $P$ is positive or zero. The absolute value equation $|x|=P$ is simply a compound equation in disguise:

$|x| = P \qquad \Longleftrightarrow \qquad x=P \quad \mbox{or} \quad x = -P$

To solve an absolute value equation:

1. Use algebra to write the equation in the form $|X|=P$. In this context, $X$ and $P$ may be entire algebraic expressions, but the equation can only make sense if $P \ge 0$.
2. Solve the compound equation $X=P \mbox{ or } X =-P$.

#### Examples

Solve for $x$: $\quad |2x+4|=8$

Solve for $w$: $\quad -5|w-7|+2 = -13$

Solve for $y$: $\quad \displaystyle \left|5-\frac{2}{3} y \right| -9 = 8$

Solve for $x$: $\quad |2x-7|=0$

Solve for $x$: $\quad |2x-7|=-5$

Solve for $x$: $\quad |2x+7| = |x-1|$

### Absolute Value Inequalities

Just as we did with absolute value equations, we'll solve absolute value inequalities by rewriting them as compound inequalities. There are two cases to consider (assuming $P>0$):

• $|x| < P \qquad \Longleftrightarrow \qquad -P < x < P$
• $|x| > P \qquad \Longleftrightarrow \qquad x > P \quad \mbox{or} \quad x < -P$

These ideas should make sense if you think about $x$ representing a distance from zero.

Important idea: Be on the lookout for equations and inequalities that are never true or always true. Usually you can spot these before you even start the solution process.

#### Examples

Solve for $x$: $\quad \displaystyle \frac{|3x+2|}{4} \le 1$

Solve for $r$: $\quad -5|r-4|+12 > -28$

Solve for $x$: $\quad |3x-2| < -5$

Solve for $t$: $\quad \displaystyle \left| 3+ \frac{t}{2} \right| > 6$

Solve for $x$: $\quad |3x+7|-2 > -5$