Solve |5x - 2| ≤ 3x + 4: Absolute Value Inequality Challenge

To solve the problem, we'll follow these steps:

• Step 1: Solve the inequality $\left|5x - 2\right| \leq 3x + 4$ by considering two cases.
• Step 2: Combine the solutions of both cases to find the intersection of permissible values for $x$.

Step 1: Consider the inequality $- (3x + 4) \leq 5x - 2 \leq 3x + 4$.
Case 1: Solve $5x - 2 \geq -(3x + 4)$, which simplifies to:
$5x - 2 \geq -3x - 4$
Add $3x$ to both sides:
$8x - 2 \geq -4$
Add 2 to both sides:
$8x \geq -2$
Divide by 8:
$x \geq -\frac{1}{4}$

Step 2: Now solve $5x - 2 \leq 3x + 4$ to get:
Subtract $3x$ from both sides:
$2x - 2 \leq 4$
Add 2 to both sides:
$2x \leq 6$
Divide by 2:
$x \leq 3$

Step 3: Combine solutions from Case 1 and Case 2.
$-\frac{1}{4} \leq x \leq 3$

Therefore, the solution to the problem is: $-\frac{1}{4} \le x \le 3$.