For what range of values of x is $\frac{1}{2}$x + $\frac{1}{4}$ > $\frac{1}{3}$x + $\frac{1}{2}$?

Analyze the question: We need to find all values of x that make the inequality (1/2)x + 1/4 > (1/3)x + 1/2 true.

Concept: Treat an inequality like an equation. Collect the x-terms on one side and the numbers on the other. Only flip the inequality sign if you multiply or divide by a negative number.

Solution:

1. Start with: (1/2)x + 1/4 > (1/3)x + 1/2

2. Subtract (1/3)x from both sides and subtract 1/4 from both sides: (1/2)x − (1/3)x > 1/2 − 1/4

3. Simplify each side: Left: 1/2 − 1/3 = 3/6 − 2/6 = 1/6, so the left side is (1/6)x. Right: 1/2 − 1/4 = 1/4.

(1/6)x > 1/4

4. Multiply both sides by 6 (positive, so the sign stays the same): x > 6/4 = 3/2

Check: Try x = 2. Left: 1 + 0.25 = 1.25. Right: 0.667 + 0.5 = 1.167. Since 1.25 > 1.167, it works. Try x = 0: left is 0.25 and right is 0.5, so it fails. This agrees with x > 3/2.

Answer: x > 3/2, which is option B.

Tip: To avoid fractions, multiply the whole inequality by the LCM of the denominators (here 12): 6x + 3 > 4x + 6, so 2x > 3, so x > 3/2. Remember that the sign only flips when you multiply or divide by a negative number.