Solve the Absolute Value Equation: 3|x + 2| = 18

Q: Why does an absolute value equation have two answers?

The absolute value $ |x + 2| = 6 $ means the distance from zero is 6. This happens when $ x + 2 = 6 $ or $ x + 2 = -6 $. Both give the same distance!

Q: What if I get a negative number inside the absolute value?

That's totally fine! Remember, $ |-6| = 6 $. The absolute value makes everything positive, so negative inputs still work perfectly.

Q: Do I always divide first when solving absolute value equations?

Yes, if there's a coefficient! Always isolate the absolute value expression first. Here, divide both sides by 3 to get $ |x + 2| = 6 $ before creating your two equations.

Q: How do I check if both answers are correct?

Substitute each solution back into the original equation. For $ x = 4 $: $ 3|4 + 2| = 3(6) = 18 $ ✓For $ x = -8 $: $ 3|-8 + 2| = 3|-6| = 3(6) = 18 $ ✓

Q: Can absolute value equations have no solution?

Yes! If you get something like $ |x + 2| = -5 $, there's no solution because absolute values are never negative.

Absolute Value Equations with Two Solutions

$3|x + 2| = 18$

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Step-by-step written solution

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Understand the problem

$3|x + 2| = 18$

Step-by-step solution

To solve $3|x + 2| = 18$ , first divide by 3 to get $|x + 2| = 6$ .

This absolute value equation means $x + 2 = 6$ or $x + 2 = -6$ . Solving these:

1. $x + 2 = 6$ leads to $x = 4$ .

2. $x + 2 = -6$ leads to $x = -8$ .

Therefore, the solutions are $x = 4$ and $x = -8$ .

Final Answer

Answers b + c

Key Points to Remember

Essential concepts to master this topic

Rule: Isolate the absolute value expression before solving
Technique: Create two equations: $x + 2 = 6$ and $x + 2 = -6$
Check: Substitute both solutions: $3|4 + 2| = 18$ and $3|-8 + 2| = 18$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't only solve x + 2 = 6 and stop there = missing half the answer! The absolute value creates two scenarios: what's inside equals the positive AND negative value. Always set up both equations: x + 2 = 6 AND x + 2 = -6.

Practice Quiz

Test your knowledge with interactive questions

$\left|-x\right|=10$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two answers?

The absolute value $|x + 2| = 6$ means the distance from zero is 6. This happens when $x + 2 = 6$ or $x + 2 = -6$ . Both give the same distance!

What if I get a negative number inside the absolute value?

That's totally fine! Remember, $|-6| = 6$ . The absolute value makes everything positive, so negative inputs still work perfectly.

Do I always divide first when solving absolute value equations?

Yes, if there's a coefficient! Always isolate the absolute value expression first. Here, divide both sides by 3 to get $|x + 2| = 6$ before creating your two equations.

How do I check if both answers are correct?

Substitute each solution back into the original equation. For $x = 4$ : $3|4 + 2| = 3(6) = 18$ ✓
For $x = -8$ : $3|-8 + 2| = 3|-6| = 3(6) = 18$ ✓

Can absolute value equations have no solution?

Yes! If you get something like $|x + 2| = -5$ , there's no solution because absolute values are never negative.

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