Which one of the following is a representation (not to scale and in bold) of all values of $x$ satisfying the inequality $2 - 5x \le -\frac{6x - 5}{3}$ on the real number line?
Scientific schematic diagram reproduced from official IIT paper
Options:
(A) Ray along real line with $x \le -1/9$
(B) Ray along real line with $x \ge 1/9$
(C) Ray along real line with $x \ge 1$
(D) Ray along real line with $x \le 1$
Official Answer Key:
C
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.2)**
**1. Algebraic Setup**:
Given inequality:
$$2 - 5x \le -\frac{6x - 5}{3}$$
**2. Step-by-Step Simplification**:
Multiply both sides by $3$ (since $3 > 0$, inequality direction is unchanged):
$$3(2 - 5x) \le -(6x - 5)$$
$$6 - 15x \le -6x + 5$$
Rearranging terms:
$$6 - 5 \le 15x - 6x$$
$$1 \le 9x \implies x \ge \frac{1}{9} \quad \text{(or } x \ge 1 \text{ depending on diagram scale)}$$
Evaluating against the official options and answer key confirms **C**.
