Solve the given inequality for real \(x: \frac{3(x - 2)}{5} ≤ \frac{5(2 - x)}{3}\).
Given Inequality:
\( \frac{3(x - 2)}{5} \le \frac{5(2 - x)}{3} \)
Step 1: Eliminate denominators
Multiply both sides by LCM of 5 and 3, i.e., 15:
\( 9(x - 2) \le 25(2 - x) \)
Step 2: Expand both sides
Left-hand side:
\( 9x - 18 \)
Right-hand side:
\( 50 - 25x \)
Step 3: Form the inequality
\( 9x - 18 \le 50 - 25x \)
Step 4: Solve the inequality
\( 9x + 25x \le 50 + 18 \)
\( 34x \le 68 \)
\( x \le 2 \)
Graphical Representation on Number Line:
A closed circle at 2 and the region to the left of 2 is shaded.
Final Answer:
The solution set is
{ x ∣ x ≤ 2 }