Let’s review the problem carefully.

The equation is:

$|x - 1| = 7x - 13$

When solving absolute value equations like this, there are two cases to consider because the absolute value of an expression can either be equal to a positive or a negative counterpart.

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Case 1: $x - 1 = 7x - 13$

This is the case where the absolute value equals the expression as-is.

1. Simplify: $x - 1 = 7x - 13$
2. Subtract $x$ from both sides: $-1 = 6x - 13$
3. Add 13 to both sides: $12 = 6x$
4. Divide by 6: $x = 2$

This solution is correct.

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Case 2: $x - 1 = -(7x - 13)$

This is the case where the absolute value equals the negative of the expression.

1. Simplify the negative sign: $x - 1 = -7x + 13$
2. Add $7x$ to both sides: $8x - 1 = 13$
3. Add 1 to both sides: $8x = 14$
4. Divide by 8: $x = \frac{7}{4}$

This solution is also correct.

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Checking Your Work

1. Check $x = 2$: Substitute $x = 2$ into the original equation: $|2 - 1| = 7(2) - 13$ $1 = 14 - 13$ $1 = 1 \quad \text{(Valid solution).}$

2. Check $x = \frac{7}{4}$: Substitute $x = \frac{7}{4}$ into the original equation: $\left|\frac{7}{4} - 1\right| = 7\left(\frac{7}{4}\right) - 13$ Simplify the left-hand side: $\left|\frac{7}{4} - \frac{4}{4}\right| = \left|\frac{3}{4}\right| = \frac{3}{4}$ Simplify the right-hand side: $7 \cdot \frac{7}{4} - 13 = \frac{49}{4} - \frac{52}{4} = -\frac{3}{4}$ The absolute value doesn’t match because $x = \frac{7}{4}$ causes the left and right sides to be unequal.

   Thus, $x = \frac{7}{4}$ is extraneous.

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Error in Your Work

The issue lies in not verifying both solutions. While solving absolute value equations, it’s essential to plug the solutions back into the original equation to check for extraneous results.

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Would you like help with more examples of checking extraneous solutions, or explanations of why they arise?

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5 Related Questions:

1. What are common strategies to avoid extraneous solutions in absolute value equations?
2. How do you solve equations with two absolute values (e.g., $|x - 3| = |2x + 1|$)?
3. What are the graphical interpretations of absolute value equations like this?
4. Can inequalities with absolute values be solved similarly?
5. How do absolute value equations relate to piecewise functions?

Tip:

Always verify your solutions by substituting them back into the original equation to catch extraneous roots!