The GATE Grind

GATE 2022 ME (ME2) – Question 2

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

Which one of the following is a representation (not to scale and in bold) of all values of $x$ satisfying the inequality $2-5x\leq-\frac{6x-5}{3}$ on the real number line?

four number lines with closed endpoints and bold rays.
  1. Option A in the figure
  2. Option B in the figure
  3. Option C in the figure
  4. Option D in the figure

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (C) Option C in the figure

Explanation

**Step 1: clear the fraction.** Multiply both sides by 3:
$$3(2-5x)\leq-(6x-5)\;\Rightarrow\;6-15x\leq-6x+5.$$

**Step 2: collect the terms in $x$.**
$$6-5\leq 15x-6x\;\Rightarrow\;1\leq 9x\;\Rightarrow\;x\geq\frac19.$$

**Step 3: read the number line.** The solution set is every $x$ greater than or equal to $1/9$. The endpoint $1/9$ is included, so it must be a closed dot, and the bold part must be the ray running to the right of a small positive value.

Only option **C** shows a closed endpoint at a positive value with the ray going right.