Solve $\frac{1}{3}$(5 - 3x) < $\frac{2}{5}$(3 - 7x)

Step 1: Analyze the question We need to solve a linear inequality and find the range of x. Fractions are involved, so we clear them first.

Step 2: Concept Multiply both sides by the LCM of the denominators to remove fractions. If you multiply or divide by a negative number, the inequality sign flips.

Step 3: Solution

(1/3)(5 − 3x) < (2/5)(3 − 7x)

The LCM of 3 and 5 is 15. Multiply both sides by 15 (positive, so the sign stays):

5(5 − 3x) < 6(3 − 7x)

25 − 15x < 18 − 42x

Add 42x to both sides:

25 + 27x < 18

Subtract 25:

27x < −7

x < −7/27

Step 4: Answer x < −7/27, which matches option D.

Tip: When there are fractions on both sides, multiply by the LCM first. This avoids messy fraction arithmetic. Always check whether you divided by a negative number, because that is the only time the sign flips. Here we divided by +27, so the sign stayed the same.

Quick check: Try x = −1 (which is less than −7/27):

• Left: (1/3)(5 + 3) = 8/3 ≈ 2.67
• Right: (2/5)(3 + 7) = 4
• 2.67 < 4 ✓