Solve the following inequality and graph the solution set. \( |-4 x+2| \leq 1 \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is (Type your answer in interval notation. Simplify your answer. Use integers or fractions for any nut B. The solution set is \( \varnothing \).

To solve the inequality \( |-4x + 2| \leq 1 \), we can break it down into two separate inequalities: 1. \( -4x + 2 \leq 1 \) 2. \( -4x + 2 \geq -1 \) For the first inequality: \[ -4x + 2 \leq 1 \implies -4x \leq -1 \implies x \geq \frac{1}{4} \] For the second inequality: \[ -4x + 2 \geq -1 \implies -4x \geq -3 \implies x \leq \frac{3}{4} \] Combining these two results gives us the interval: \[ \frac{1}{4} \leq x \leq \frac{3}{4} \] In interval notation, the solution set is: \[ \left[\frac{1}{4}, \frac{3}{4}\right] \] So, the correct choice is: A. The solution set is \(\left[\frac{1}{4}, \frac{3}{4}\right]\).