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 should be 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 $X=P \mbox{ or } X =-P$.
3. Check your answers.

Examples

• [1] Solve for $x$: $\quad |2x+4|=8$ (Solution)

• [2] Solve for $w$: $\quad -5|w-7|+2 = -13$ (Solution)

• [3] Solve for $y$: $\quad \displaystyle \left|5-\frac{2}{3} y \right| -9 = 8$ (Solution)

• [4] Solve for $x$: $\quad |2x-7|=0$ (Solution)

• [5] Solve for $x$: $\quad |2x-7|=-5$ (Solution)

• [6] Solve for $x$: $\quad |2x+7| = |x-1|$ (Solution)

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

• [7] Solve for $x$: $\quad \displaystyle \frac{|3x+2|}{4} \le 1$ (Solution)

• [8] Solve for $r$: $\quad -5|r-4|+12 > -28$ (Solution)

• [9] Solve for $x$: $\quad |3x-2| < -5$ (Solution)

• [10] Solve for $t$: $\quad \displaystyle \left| 3+ \frac{t}{2} \right| > 6$ (Solution)

• [11] Solve for $x$: $\quad |3x+7|-2 > -5$ (Solution)

For further help...

1. Khan Academy has excellent videos on solving absolute value inequalities. Start at this page.
2. Do a Google search for "solving absolute value equations".
3. Do a Google search for "solving absolute value inequalities".