Absolute value equations
Introduction
The absolute value of a number is its distance from zero on the number line. Since distance is never negative, the absolute value is never negative. The absolute value of a number $A$ is written $|A|$.
An equation that contains one or more absolute values is called an absolute value equation. For example, $|2x-3|=4$, $|4y|=-7$, and $|u-8|=|2u+1|$ are absolute value equations.
Solving absolute value equation
If $|X|=a$, then $X=a$ or $X=-a$, provided $a$ is positive or zero.
If $a$ is negative, then $|X|=a$ has no solution, because an absolute value can never be negative.
If the absolute value is not by itself, isolate it first, then apply the rule above.
Example 1: Solve $|2x-4|=5$.
Solution: Since $5$ is positive,
$2x-4=5$ or $2x-4=-5$
Solve each equation. For $2x-4=5$:
$\begin{align*}2x&=9\\x&=\dfrac{9}{2}\end{align*}$
For $2x-4=-5$:
$\begin{align*}2x&=-1\\x&=-\dfrac{1}{2}\end{align*}$
The solutions are $\left\{\dfrac{9}{2},\,-\dfrac{1}{2}\right\}$.
Example 2: Solve $|2y-5|+6=2$.
Solution: First isolate the absolute value by subtracting $6$ from both sides:
$\begin{align*}|2y-5|&=2-6\\|2y-5|&=-4\end{align*}$
The absolute value equals a negative number, so there is no solution; that is, the solution is $\emptyset$.
Example 3: Solve $3|2u-3|-7=56$.
Solution: First isolate the absolute value. Add $7$ to both sides, then divide by $3$:
$\begin{align*}3|2u-3|&=63\\|2u-3|&=\dfrac{63}{3}\\|2u-3|&=21\end{align*}$
Since $21$ is positive,
$2u-3=21$ or $2u-3=-21$
Solve each equation. For $2u-3=21$:
$\begin{align*}2u&=24\\u&=12\end{align*}$
For $2u-3=-21$:
$\begin{align*}2u&=-18\\u&=-9\end{align*}$
The solutions are $\{12,\,-9\}$.
Equality of absolute values
If $|X|=|Y|$, then $X=Y$ or $X=-Y$.
Example 1: Solve $|2w-13|=|5-7w|$.
Solution: The two absolute values are equal, so
$2w-13=5-7w$ or $2w-13=-(5-7w)$
Solve the first equation:
$\begin{align*}2w-13&=5-7w\\2w+7w&=5+13\\9w&=18\\w&=2\end{align*}$
Solve the second equation:
$\begin{align*}2w-13&=-(5-7w)\\2w-13&=-5+7w\\2w-7w&=-5+13\\-5w&=8\\w&=-\dfrac{8}{5}\end{align*}$
The solutions are $\left\{2,\,-\dfrac{8}{5}\right\}$.
Example 2: Solve $|3x-10|=|3x+8|$.
Solution: The two absolute values are equal, so
$3x-10=3x+8$ or $3x-10=-(3x+8)$
For the first equation, subtracting $3x$ from both sides gives $-10=8$, which is false, so it has no solution.
Solve the second equation:
$\begin{align*}3x-10&=-(3x+8)\\3x-10&=-3x-8\\3x+3x&=-8+10\\6x&=2\\x&=\dfrac{1}{3}\end{align*}$
The solution is $\left\{\dfrac{1}{3}\right\}$.
Example 3: Solve $\left|\dfrac{1}{3}m-4\right|=\left|\dfrac{5}{6}m-2\right|$.
Solution: The two absolute values are equal, so
$\dfrac{1}{3}m-4=\dfrac{5}{6}m-2$ or $\dfrac{1}{3}m-4=-\left(\dfrac{5}{6}m-2\right)$
The denominators are $3$ and $6$, so the least common denominator is $6$. Multiply each equation by $6$ to clear the fractions. For the first equation:
$\begin{align*}2m-24&=5m-12\\2m-5m&=-12+24\\-3m&=12\\m&=-4\end{align*}$
For the second equation:
$\begin{align*}2m-24&=-5m+12\\2m+5m&=12+24\\7m&=36\\m&=\dfrac{36}{7}\end{align*}$
The solutions are $\left\{-4,\,\dfrac{36}{7}\right\}$.