Solve the inequality for real x: (3(x - 2))/5 ≤ (5(2 - x))/3

Question 11 of 40intermediate🔧 ApplyNumerical2 marks

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

Correct Answer

The solution set is $(-\infty, 2]$ .

Exercise: EXERCISE 5.1 | Q: 11 | (Page Page 7)

For More Understanding

Explanation

This question tests the student's ability to solve linear inequalities with fractions. Following the method shown in Example 4 of the textbook, we first eliminate denominators by multiplying both sides by the LCM (15), then collect like terms and solve for x. The key rule to remember is Rule 2: when multiplying or dividing by a positive number, the inequality sign remains unchanged.

Solution Steps

Step 1: Multiply both sides by 15 (LCM of 5 and 3): $9(x-2) \leq 25(2-x)$
Step 2: Expand both sides: $9x - 18 \leq 50 - 25x$
Step 3: Collect x terms on one side: $9x + 25x \leq 50 + 18$ , which gives $34x \leq 68$
Step 4: Divide by 34 (positive number): $x \leq 2$
Step 5: Solution set is $x \in (-\infty, 2]$

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