Solve the Absolute Value Equation: |3x-5| = 12

Q: Why do I need to solve two equations for one absolute value?

Because absolute value measures distance from zero! The expression $3x-5$ could be 12 units away in either direction: +12 or -12.

Q: How do I convert improper fractions to mixed numbers?

Divide the numerator by the denominator! For $\frac{17}{3}$: 17 ÷ 3 = 5 remainder 2, so $5\frac{2}{3}$. For $\frac{-7}{3}$: -7 ÷ 3 = -2 remainder -1, so $-2\frac{1}{3}$.

Q: Do I always get exactly two solutions?

Usually yes, but not always! You get two solutions when both equations give valid answers. Sometimes one equation might give an answer that doesn't work when you check it.

Q: How can I check my mixed number answers easily?

Convert back to improper fractions for easier calculation! $5\frac{2}{3} = \frac{17}{3}$ and $-2\frac{1}{3} = \frac{-7}{3}$. Then substitute and verify both give 12.

Absolute Value Equations with Mixed Number Solutions

$\left|3x-5\right|=12$

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Step-by-step written solution

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Understand the problem

$\left|3x-5\right|=12$

Step-by-step solution

To solve the equation $\left|3x-5\right|=12$ , we split it into two separate equations:

1. $3x-5=12$

2. $3x-5=-12$

For the first equation:

$3x-5=12$

Add 5 to both sides:

$3x=17$

Divide both sides by 3:

$x=5\frac{2}{3}$

For the second equation:

$3x-5=-12$

Add 5 to both sides:

$3x=-7$

Divide both sides by 3:

$x=-2\frac{1}{3}$

Thus, the solutions are $x=5\frac{2}{3}$ and $x=-2\frac{1}{3}$ .

Final Answer

$x=5\frac{2}{3}$ , $x=-2\frac{1}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations split into two separate linear equations
Technique: If |expression| = 12, then expression = 12 or expression = -12
Check: Substitute both answers: |3(17/3)-5| = 12 and |3(-7/3)-5| = 12 ✓

Common Mistakes

Avoid these frequent errors

Only solving one equation instead of both
Don't just solve 3x - 5 = 12 and stop = only one solution instead of two! Absolute value means the expression inside could be positive OR negative. Always create both equations: 3x - 5 = 12 AND 3x - 5 = -12.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do I need to solve two equations for one absolute value?

Because absolute value measures distance from zero! The expression $3x-5$ could be 12 units away in either direction: +12 or -12.

How do I convert improper fractions to mixed numbers?

Divide the numerator by the denominator! For $\frac{17}{3}$ : 17 ÷ 3 = 5 remainder 2, so $5\frac{2}{3}$ . For $\frac{-7}{3}$ : -7 ÷ 3 = -2 remainder -1, so $-2\frac{1}{3}$ .

What if I get the same answer for both equations?

That's rare but possible! It happens when the expression inside the absolute value bars equals zero. You'd still have a valid solution, just one solution instead of two.

Do I always get exactly two solutions?

Usually yes, but not always! You get two solutions when both equations give valid answers. Sometimes one equation might give an answer that doesn't work when you check it.

How can I check my mixed number answers easily?

Convert back to improper fractions for easier calculation! $5\frac{2}{3} = \frac{17}{3}$ and $-2\frac{1}{3} = \frac{-7}{3}$ . Then substitute and verify both give 12.

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