Question

Solve the given inequality for real \( x \): \[ \frac{3(x - 2)}{5} \leq \frac{5(2 - x)}{3} \]

Hint:

To solve inequalities involving fractions, clear the fractions by multiplying both sides by the least common multiple of the denominators.

Answer 1

Step 1: Eliminate the fractions.
Multiply both sides of the inequality by 15, the least common multiple of 5 and 3, to eliminate the denominators: \[ 15 \times \frac{3(x - 2)}{5} \leq 15 \times \frac{5(2 - x)}{3} \] Simplifying: \[ 3 \times 3(x - 2) \leq 5 \times 5(2 - x) \] \[ 9(x - 2) \leq 25(2 - x) \]
Step 2: Expand both sides.
\[ 9x - 18 \leq 50 - 25x \]
Step 3: Move all terms involving \( x \) to one side and constants to the other side.
\[ 9x + 25x \leq 50 + 18 \] \[ 34x \leq 68 \]
Step 4: Solve for \( x \).
\[ x \leq \frac{68}{34} = 2 \] So, the solution is \( x \leq 2 \).