Solve the Absolute Value Equation: |2x - 4| = 8

Q: Why do absolute value equations have two solutions?

Because the absolute value represents distance from zero, which can come from both positive and negative directions. For example, both 8 and -8 are distance 8 from zero!

Q: How do I know when to make the right side negative?

You always create two cases: one where the expression inside equals the positive value, and one where it equals the negative value. So $ |2x - 4| = 8 $ becomes both $ 2x - 4 = 8 $ and $ 2x - 4 = -8 $.

Q: Can absolute value equations have no solutions?

Yes! If the right side is negative (like $ |x| = -5 $), there are no solutions because absolute values are never negative.

Q: Do I solve both cases the same way?

Yes! Once you set up the two cases, solve each one like a regular linear equation. Add, subtract, multiply, and divide normally to isolate x.

Absolute Value Equations with Two Solutions

$\left|2x - 4\right|=8$

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Step-by-step written solution

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Understand the problem

$\left|2x - 4\right|=8$

Step-by-step solution

To solve $\left|2x - 4\right|=8$ , we split it into two cases based on the nature of absolute values:

1) $2x - 4=8$ :

Add $4$ to both sides: $2x=12$ .

Divide both sides by $2$ to get $x=6$ .

2) $2x - 4=-8$ :

Add $4$ to both sides: $2x=-4$ .

Divide both sides by $2$ to get $x=-2$ .

So, the solutions are $x=-2$ and $x=6$ .

Final Answer

$x=-2$ , $x=6$

Key Points to Remember

Essential concepts to master this topic

Definition: Absolute value equation splits into two separate linear cases
Method: Set 2x - 4 = 8 and 2x - 4 = -8
Verify: Check both solutions: |2(6) - 4| = 8 and |2(-2) - 4| = 8 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't just solve 2x - 4 = 8 and stop there = missing half the solutions! Absolute value means the expression inside could be positive OR negative. Always set up both cases: positive and negative.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why do absolute value equations have two solutions?

Because the absolute value represents distance from zero, which can come from both positive and negative directions. For example, both 8 and -8 are distance 8 from zero!

How do I know when to make the right side negative?

You always create two cases: one where the expression inside equals the positive value, and one where it equals the negative value. So $|2x - 4| = 8$ becomes both $2x - 4 = 8$ and $2x - 4 = -8$ .

What if I only get one solution that works?

That can happen! Sometimes one of the solutions doesn't satisfy the original equation when you check. Always verify both by substituting back into the original absolute value equation.

Can absolute value equations have no solutions?

Yes! If the right side is negative (like $|x| = -5$ ), there are no solutions because absolute values are never negative.

Do I solve both cases the same way?

Yes! Once you set up the two cases, solve each one like a regular linear equation. Add, subtract, multiply, and divide normally to isolate x.

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