The given equation is:

$|3x - 4| = x + 2$

To solve this, we handle the absolute value equation by considering the two cases for $|3x - 4|$:

Case 1: $3x - 4 = x + 2$

1. Solve for $x$: $3x - x = 2 + 4$ $2x = 6 \implies x = 3$

Case 2: $3x - 4 = -(x + 2)$

1. Simplify: $3x - 4 = -x - 2$ $3x + x = -2 + 4$ $4x = 2 \implies x = \frac{1}{2}$

Verify Both Solutions

To verify:

1. For $x = 3$: $|3(3) - 4| = |9 - 4| = 5, \quad 3 + 2 = 5 \quad \text{(valid)}.$

2. For $x = \frac{1}{2}$: $|3\left(\frac{1}{2}\right) - 4| = |1.5 - 4| = 2.5, \quad \frac{1}{2} + 2 = 2.5 \quad \text{(valid)}.$

Final Answer:

The solutions are: $x = 3 \quad \text{and} \quad x = \frac{1}{2}.$

Would you like a detailed explanation of any step?

Related Questions:

1. How do you verify absolute value equations graphically?
2. What is the general method for solving absolute value equations?
3. How do you determine whether an absolute value equation has no solution?
4. Can absolute value equations result in inequalities? If so, how?
5. How do you extend this solution method to systems of equations?

Tip:

Always verify potential solutions for absolute value equations to ensure they satisfy the original equation, as extraneous solutions may arise.