Solve the Absolute Value Equation: |6x-12| = 6

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

Because absolute value measures distance from zero, which is always positive. So $ |6x-12|=6 $ means the expression inside could be 6 units away in either direction: +6 or -6.

Q: How do I know which case is positive and which is negative?

Set up both cases: Case 1: $ 6x-12=6 $ (expression equals positive 6) and Case 2: $ -(6x-12)=6 $ (expression equals negative 6, so we negate it).

Q: Do I always get exactly two solutions?

Not always! You can get two solutions (like this problem), one solution, or no solutions depending on the equation. Always solve both cases to find out.

Q: How can I check if my solutions are correct?

Substitute each solution back into the original equation. For x=1: $ |6(1)-12|=|-6|=6 $ ✓. For x=3: $ |6(3)-12|=|6|=6 $ ✓.

Absolute Value Equations with Two Solutions

$\left|6x-12\right|=6$

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Understand the problem

$\left|6x-12\right|=6$

Step-by-step solution

To solve this exercise, we need to note that the left side is in absolute value.

Absolute value checks the distance of a number from zero, meaning its solution is always positive.

Therefore, we have two possibilities, either the numbers inside will be positive or negative,

In other words, we check two options, in one what's inside the absolute value is positive and in the second it's negative.

6x-12=6

6x=18

x=3

This is the first solution

-(6x-12)=6
-6x+12=6
-6x=6-12
-6x=-6
6x=6
x=1

And this is the second solution,

So we found two solutions,

x=1, x=3

And that's the solution!

Final Answer

$x=1$ , $x=3$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equals positive number creates two cases
Technique: Solve 6x-12=6 and -(6x-12)=6 to get x=3,1
Check: Substitute both solutions: |6(1)-12|=6 and |6(3)-12|=6 ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't solve only 6x-12=6 to get just x=3! This misses half the solutions because absolute value creates two equal cases. Always set up both the positive case AND the negative case: -(6x-12)=6.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=3$

FAQ

Everything you need to know about this question

Why does an absolute value equation have two solutions?

Because absolute value measures distance from zero, which is always positive. So $|6x-12|=6$ means the expression inside could be 6 units away in either direction: +6 or -6.

How do I know which case is positive and which is negative?

Set up both cases: Case 1: $6x-12=6$ (expression equals positive 6) and Case 2: $-(6x-12)=6$ (expression equals negative 6, so we negate it).

What if I get the same answer for both cases?

That's possible! Sometimes absolute value equations have only one unique solution. This happens when the expression inside the absolute value equals zero at the solution.

Do I always get exactly two solutions?

Not always! You can get two solutions (like this problem), one solution, or no solutions depending on the equation. Always solve both cases to find out.

How can I check if my solutions are correct?

Substitute each solution back into the original equation. For x=1: $|6(1)-12|=|-6|=6$ ✓. For x=3: $|6(3)-12|=|6|=6$ ✓.

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