To solve the equation $|5x - 2| = |3x - 4|$, we need to consider the different cases where the absolute values affect the equation.

Case 1: Both expressions are positive or equal

This case assumes that both $5x - 2$ and $3x - 4$ are either both positive or both negative: $5x - 2 = 3x - 4$ Solving for $x$: $5x - 3x = -4 + 2$ $2x = -2$ $x = -1$

Case 2: One expression is positive and the other negative

This case assumes that one of the expressions is positive and the other is negative. There are two sub-cases:

Sub-case 2.1: $5x - 2$ is positive and $3x - 4$ is negative

$5x - 2 = -(3x - 4)$ Simplifying: $5x - 2 = -3x + 4$ $5x + 3x = 4 + 2$ $8x = 6$ $x = \frac{6}{8} = \frac{3}{4}$

Sub-case 2.2: $5x - 2$ is negative and $3x - 4$ is positive

$-(5x - 2) = 3x - 4$ Simplifying: $-5x + 2 = 3x - 4$ $-5x - 3x = -4 - 2$ $-8x = -6$ $x = \frac{-6}{-8} = \frac{3}{4}$

Solution Set

The two possible solutions are $x = -1$ and $x = \frac{3}{4}$.

Verification

We should verify these solutions by substituting them back into the original equation.

• For $x = -1$: $|5(-1) - 2| = |3(-1) - 4|$ $|-5 - 2| = |-3 - 4|$ $|-7| = |-7| \quad \text{(True)}$

• For $x = \frac{3}{4}$: $|5\left(\frac{3}{4}\right) - 2| = |3\left(\frac{3}{4}\right) - 4|$ $\left| \frac{15}{4} - \frac{8}{4} \right| = \left| \frac{9}{4} - \frac{16}{4} \right|$ $\left| \frac{7}{4} \right| = \left| \frac{-7}{4} \right| \quad \text{(True)}$

Thus, both solutions $x = -1$ and $x = \frac{3}{4}$ are valid.

Would you like more details or have any questions?

Related Questions:

1. How do you solve absolute value inequalities?
2. What are the key differences between solving absolute value equations and inequalities?
3. Can you graph the equation $|5x - 2| = |3x - 4|$?
4. How does the concept of absolute value apply to real-world situations?
5. What are some common mistakes to avoid when solving absolute value equations?

Tip:

Always verify your solutions when solving absolute value equations, as some solutions might not satisfy the original equation after accounting for the absolute values.