To solve absolute value equations, follow these general steps:
Steps to Solve Absolute Value Equations
- Isolate the absolute value expression on one side of the equation.
For example, if you have ∣2x−1∣+3=6|2x-1|+3=6∣2x−1∣+3=6, first isolate the absolute value: ∣2x−1∣=3|2x-1|=3∣2x−1∣=3
- Check the number on the other side of the equation.
- If it is negative, there is no solution because absolute values cannot be negative
* If it is zero or positive, proceed to the next step.
- Set up two separate equations by removing the absolute value bars and considering both the positive and negative cases:
If ∣A∣=b, then A=b or A=−b\text{If }|A|=b,\text{ then }A=b\text{ or }A=-bIf ∣A∣=b, then A=b or A=−b
For example, if ∣x+2∣=3|x+2|=3∣x+2∣=3, then
x+2=3x+2=3x+2=3 or x+2=−3x+2=-3x+2=−3
- Solve each resulting equation as a normal linear equation.
For the example above:
x=1x=1x=1 or x=−5x=-5x=−5
- Check your solutions by substituting them back into the original equation to verify they satisfy it
Additional Notes
- If the equation contains absolute values on both sides, set the expressions equal to each other and also set one equal to the negative of the other, then solve both equations
.
For example, ∣ax+b∣=∣cx+d∣|ax+b|=|cx+d|∣ax+b∣=∣cx+d∣ leads to:
ax+b=cx+dorax+b=−(cx+d)ax+b=cx+d\quad \text{or}\quad ax+b=-(cx+d)ax+b=cx+dorax+b=−(cx+d)
- If the absolute value is multiplied by a negative coefficient, first divide both sides by that negative number to make the absolute value positive before proceeding
- If the absolute value equals zero, solve the equation inside the absolute value directly since ∣A∣=0 ⟹ A=0|A|=0\implies A=0∣A∣=0⟹A=0
Summary Example
Solve ∣x−5∣=3|x-5|=3∣x−5∣=3:
-
Write two equations:
x−5=3x-5=3x−5=3 or x−5=−3x-5=-3x−5=−3 -
Solve each:
x=8x=8x=8 or x=2x=2x=2 -
These are the solutions
This approach works for all basic absolute value equations and can be extended to more complex cases by isolating the absolute value terms first and then applying the same principle
