Solving Absolute Value Equation: |-x+6| = |3x-2|

Q: Why do I need to solve two separate equations?

Because absolute value means distance from zero, which is always positive! The expression inside could be positive or negative, so we need to consider both possibilities to find all solutions.

Q: How do I set up the second case correctly?

For the second case, set the left side equal to the negative of the right side. So $|-x + 6| = |3x - 2|$ becomes $-x + 6 = -(3x - 2)$, which simplifies to $-x + 6 = -3x + 2$.

Q: What if one of my solutions doesn't work when I check?

Sometimes absolute value equations have extraneous solutions that don't satisfy the original equation. Always check both answers - if one doesn't work, just use the valid solution(s).

Q: Do I always get exactly two solutions?

Not always! You might get two solutions (like this problem), one solution (if both cases give the same answer), or no solutions (if both fail the verification step).

Absolute Value Equations with Two Cases

$|-x+6|=|3x-2|$

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Understand the problem

$|-x+6|=|3x-2|$

Step-by-step solution

To solve the problem, follow these steps:

Case 1: Set $-x + 6 = 3x - 2$ .
Simplify and solve for $x$ :

$-x + 6 = 3x - 2$
Add $x$ to both sides: $6 = 4x - 2$
Add 2 to both sides: $8 = 4x$
Divide both sides by 4: $x = 2$

Case 2: Set $-x + 6 = -(3x - 2)$ , which is $-x + 6 = -3x + 2$ .
Simplify and solve for $x$ :

$-x + 6 = -3x + 2$
Add $3x$ to both sides: $2x + 6 = 2$
Subtract 6 from both sides: $2x = -4$
Divide both sides by 2: $x = -2$

Finally, verify that both solutions satisfy the original absolute value equation:

When $x = 2$ : $|-2 + 6| = |6 - 2| \Rightarrow |4| = |4|$ , which holds true.
When $x = -2$ : $|2 + 6| = |-6 - 2| \Rightarrow |8| = |8|$ , which holds true.

Thus, both $x = 2$ and $x = -2$ are valid solutions to the equation.

The solutions to the problem are $\boldsymbol{x=-2}$ and $\boldsymbol{x=2}$ .

Therefore, the correct answer choice is $\boldsymbol{x=-2}$ , $\boldsymbol{x=2}$ .

Final Answer

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

Key Points to Remember

Essential concepts to master this topic

Rule: Absolute value equations create two separate linear equations
Technique: Case 1: $-x + 6 = 3x - 2$ , Case 2: $-x + 6 = -(3x - 2)$
Check: Substitute both solutions back: when $x = 2$ , $|4| = |4|$ ✓

Common Mistakes

Avoid these frequent errors

Solving only one case or setting up cases incorrectly
Don't solve just $-x + 6 = 3x - 2$ and stop = you miss half the solutions! This gives only one answer when there should be two. Always set up both cases: positive and negative scenarios for the absolute value equation.

Practice Quiz

Test your knowledge with interactive questions

$\left|x\right|=5$

FAQ

Everything you need to know about this question

Why do I need to solve two separate equations?

Because absolute value means distance from zero, which is always positive! The expression inside could be positive or negative, so we need to consider both possibilities to find all solutions.

How do I set up the second case correctly?

For the second case, set the left side equal to the negative of the right side. So $|-x + 6| = |3x - 2|$ becomes $-x + 6 = -(3x - 2)$ , which simplifies to $-x + 6 = -3x + 2$ .

What if one of my solutions doesn't work when I check?

Sometimes absolute value equations have extraneous solutions that don't satisfy the original equation. Always check both answers - if one doesn't work, just use the valid solution(s).

Can absolute value equations have no solutions?

Yes! If neither case produces a valid solution when you check, then the equation has no solution. This happens when the setup leads to contradictions.

Do I always get exactly two solutions?

Not always! You might get two solutions (like this problem), one solution (if both cases give the same answer), or no solutions (if both fail the verification step).

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