Solve the Absolute Value Equation: Find a in |a - 2| = 6

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

Absolute value measures distance, which is always positive! When $ |a - 2| = 6 $, it means a is exactly 6 units away from 2. This happens at two places: 6 units to the right (8) and 6 units to the left (-4).

Q: How do I know which case to use first?

It doesn't matter! You need to solve both cases anyway. Some students prefer the positive case first: $ a - 2 = 6 $, then the negative: $ a - 2 = -6 $.

Q: Can absolute value equations have no solution?

Yes! If the absolute value equals a negative number (like $ |x| = -3 $), there's no solution because absolute values are never negative.

Q: How do I check my answers quickly?

Substitute each answer into the original equation. For $ a = 8 $: $ |8 - 2| = |6| = 6 $ ✓. For $ a = -4 $: $ |-4 - 2| = |-6| = 6 $ ✓.

Absolute Value Equations with Two Solutions

$\left| a - 2 \right| = 6$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$\left| a - 2 \right| = 6$

Step-by-step solution

To solve $\left| a - 2 \right| = 6$ , consider the cases:

1. $a - 2 = 6$ leads to $a = 8$

2. $a - 2 = -6$ leads to $a = -4$

Thus, the solutions are $a = 8$ and $a = -4$ .

Final Answer

$a = 8$ , $a = -4$

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equals positive number creates two separate equations
Technique: Solve both $a - 2 = 6$ and $a - 2 = -6$
Check: Verify both solutions: $|8 - 2| = 6$ and $|-4 - 2| = 6$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting the negative case
Don't only solve $a - 2 = 6$ to get just $a = 8$ ! This misses half the solution because absolute value creates distance in both directions. Always set up both positive and negative cases: $a - 2 = 6$ AND $a - 2 = -6$ .

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 answers?

Absolute value measures distance, which is always positive! When $|a - 2| = 6$ , it means a is exactly 6 units away from 2. This happens at two places: 6 units to the right (8) and 6 units to the left (-4).

How do I know which case to use first?

It doesn't matter! You need to solve both cases anyway. Some students prefer the positive case first: $a - 2 = 6$ , then the negative: $a - 2 = -6$ .

What if I get the same answer for both cases?

That's impossible for this type of problem! If you get identical answers, double-check your algebra. The two cases should always give different solutions when the absolute value equals a positive number.

Can absolute value equations have no solution?

Yes! If the absolute value equals a negative number (like $|x| = -3$ ), there's no solution because absolute values are never negative.

How do I check my answers quickly?

Substitute each answer into the original equation. For $a = 8$ : $|8 - 2| = |6| = 6$ ✓. For $a = -4$ : $|-4 - 2| = |-6| = 6$ ✓.

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