Solve the Absolute Value Equation: |x+2| = |x-2|

Q: Why do I need to consider two cases for absolute value equations?

The absolute value symbol means distance from zero, which is always positive. So |A| = |B| means two expressions have the same distance from zero - they could be equal (A = B) or opposite values (A = -B).

Q: What if one of my cases gives no solution?

That's completely normal! In this problem, Case 1 gives $ 2 = -2 $, which is impossible. Just move on to Case 2 - sometimes only one case has a valid solution.

Q: How do I know which case will give the real answer?

You don't know ahead of time - that's why you must solve both cases! Sometimes both work, sometimes only one works, and sometimes neither works (no solution).

Q: Can I solve this by graphing instead?

Yes! Graph $ y = |x+2| $ and $ y = |x-2| $. The x-coordinate where the graphs intersect is your solution. For this problem, they cross at x = 0.

Q: Why does Case 1 give 2 = -2?

When you solve $ x + 2 = x - 2 $, subtracting x from both sides leaves $ 2 = -2 $. This is a contradiction - it means no value of x can make this case true.

Absolute Value Equations with Case Analysis

$|x+2|=|x-2|$

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Understand the problem

$|x+2|=|x-2|$

Step-by-step solution

To solve the equation $|x+2|=|x-2|$ , we begin by considering the properties of absolute values.

The statement $|A| = |B|$ implies two cases:

Case 1: $A = B$
Case 2: $A = -B$

For our problem, consider:

Case 1: $x + 2 = x - 2$
Case 2: $x + 2 = -(x - 2)$

Let's solve each case:

Case 1: $x + 2 = x - 2$
Subtract $x$ from both sides: $x + 2 - x = x - 2 - x$ Reduce to: $2 = -2$ Since $2 \neq -2$ , this case has no solution.
Case 2: $x + 2 = -(x - 2)$
Expand the right-hand side: $x + 2 = -x + 2$ Add $x$ to both sides: $x + x + 2 = 2$ This simplifies to: $2x + 2 = 2$ Subtract 2 from both sides: $2x = 0$ Solve for $x$ : $x = 0$

Thus, the solution to $|x+2|=|x-2|$ is $x = 0$ . The correct answer is the choice: $x = 0$ .

Final Answer

$x=0$

Key Points to Remember

Essential concepts to master this topic

Rule: When |A| = |B|, consider both A = B and A = -B
Technique: For |x+2| = |x-2|, solve x+2 = x-2 and x+2 = -(x-2)
Check: Substitute x = 0: |0+2| = |0-2| becomes 2 = 2 ✓

Common Mistakes

Avoid these frequent errors

Only considering one case when solving absolute value equations
Don't solve just x+2 = x-2 and stop = missing the real solution! Case 1 gives no solution (2 = -2), but Case 2 gives the answer. Always check both A = B and A = -B when solving |A| = |B|.

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

Why do I need to consider two cases for absolute value equations?

The absolute value symbol means distance from zero, which is always positive. So |A| = |B| means two expressions have the same distance from zero - they could be equal (A = B) or opposite values (A = -B).

What if one of my cases gives no solution?

That's completely normal! In this problem, Case 1 gives $2 = -2$ , which is impossible. Just move on to Case 2 - sometimes only one case has a valid solution.

How do I know which case will give the real answer?

You don't know ahead of time - that's why you must solve both cases! Sometimes both work, sometimes only one works, and sometimes neither works (no solution).

Can I solve this by graphing instead?

Yes! Graph $y = |x+2|$ and $y = |x-2|$ . The x-coordinate where the graphs intersect is your solution. For this problem, they cross at x = 0.

Why does Case 1 give 2 = -2?

When you solve $x + 2 = x - 2$ , subtracting x from both sides leaves $2 = -2$ . This is a contradiction - it means no value of x can make this case true.

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