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College Algebra/Solving Linear Equations and Inequalities/Absolute Value Equations

Solving Linear Equations and Inequalities

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Absolute Value Equations

Absolute value equations can be solved by writing two separate equations.
The absolute value of a number xx is the distance from xx to zero on a number line, represented by ∣x∣|x|. If x≥0x\geq 0, ∣x∣=x\lvert x\rvert=x. If x<0x \lt 0, ∣−x∣=x\lvert -x\rvert=x. The absolute value is always positive, with the exception of zero:
∣3∣=3∣−3∣=3∣0∣=0|3|=3 \;\;\;\;\;\;\;\;\;\;|-3|=3 \;\;\;\;\;\;\;\;\;\; |0|=0
An absolute value equation contains an absolute value expression. An absolute value equation can have zero, one, or two solutions:
  • The equation ∣x∣=5|x|=5 has two solutions, x=5x=5 and x=−5x=-5.
  • The equation ∣x∣=0|x|=0 has one solution, x=0x=0.
  • The equation ∣x∣=−1|x|=-1 has no solutions because an absolute value cannot be negative.

To solve an absolute value equation, first isolate the absolute value expression. Then write the equation as two separate equations without the absolute value symbols. One equation represents the positive value of the expression, and the other represents the negative value. Then use properties of equality to solve each equation.

For more complicated absolute value equations, it may take more than one step to isolate the absolute value expression. Once isolated, write two equations using the expression inside the absolute value symbols. Then use properties of equality to isolate the variable in each equation.

Step-By-Step Example
Solving Absolute Value Equations
Solve the equation:
∣x−9∣=8\left|x-9\right|=8
Step 1
Write the equation as two separate equations without the absolute value symbols. Let one represent the positive value, and the other represent the negative value.
1)x−9=82)x−9=−81)\;x-9=8\;\;\;\;\;\;\;\;\;\;2)\;x-9=-8
Step 2

Solve the first equation.

Apply the addition property of equality to undo the subtraction. Add 9 to both sides of the equation.
x−9=8x−9+9=8+9x=17\begin{aligned}x-9&=8\\x-9+9&=8+9\\x&=17\end{aligned}
Step 3

Solve the second equation.

Apply the addition property of equality to undo the subtraction. Add 9 to both sides of the equation.
x−9=−8x−9+9=−8+9x=1\begin{aligned}x-9&=-8\\x-9+9&=-8+9\\x&=1\end{aligned}
Solution
The equation has two solutions: x=17x=17 and x=1x=1.
Step-By-Step Example
Solving Multistep Absolute Value Equations
Solve the equation.
2∣x+3∣−4=62|x+3|-4=6
Step 1
Isolate the absolute value expression. Apply the addition property of equality to undo the subtraction. Add 4 to both sides.
2∣x+3∣−4=62∣x+3∣−4+4=6+42∣x+3∣=10\begin{aligned}2|x+3|-4&=6\\2|x+3|-4+4&=6+4\\2|x+3|&=10\end{aligned}
Step 2
Apply the division property of equality to undo the multiplication. Divide both sides by 2.
2∣x+3∣=102∣x+3∣2=102∣x+3∣=5\begin{aligned}2|x+3|&=10\\ \frac{2|x+3|}{2}&=\frac{10}{2}\\ |x+3|&=5\end{aligned}
Step 3
Rewrite as two separate equations without the absolute value symbols.
1)x+3=52)x+3=−51)\;x+3=5\;\;\;\;\;\;\;\;\;\;2)\;x+3=-5
Step 4

Solve the first equation.

Apply the subtraction property of equality to undo the addition. Subtract 3 from both sides of the equation.
x+3=5x+3−3=5−3x=2\begin{aligned}x+3&=5\\x+3-3&=5-3\\x&=2\end{aligned}
Step 5

Solve the second equation.

Apply the subtraction property of equality to undo the addition. Subtract 3 from both sides of the equation.
x+3=−5x+3−3=−5−3x=−8\begin{aligned}x+3&=-5\\x+3-3&=-5-3\\x&=-8\end{aligned}
Solution
The equation has two solutions: x=2x=2 and x=−8x=-8.
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