GATE 2022 EC GA Q.7: solution and answer

GATE 2022 · ECGA Q.7 · 2 marks · MCQ

Consider the following inequalities.

(i) $2 x - 1 > 7$

(ii) $2 x - 9 < 1$

Which one of the following expressions below satisfies the above two inequalities?

(A)
$x \leq - 4$
(B)
$- 4 < x \leq 4$
(C)
$4 < x < 5$
(D)
$x \geq 5$

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Answer and solution

ANSWER

Official answer: CSource: Official answer key, GATE 2022 EC (IIT Kharagpur)

Inequality (i). Adding the same number to both sides keeps the inequality: $2 x - 1 > 7 \Rightarrow 2 x > 8$ . Dividing by a positive number keeps the direction, so $x > 4$ .
Inequality (ii). In the same way, $2 x - 9 < 1 \Rightarrow 2 x < 10 \Rightarrow x < 5$ .
Both must hold, so take the intersection of the two solution sets: ${x > 4} \cap {x < 5}$ gives $4 < x < 5$ . Both ends are strict because both inequalities are strict.
Check the options. (A) $x \leq - 4$ and (B) $- 4 < x \leq 4$ break (i), because every such $x$ is at most 4. (D) $x \geq 5$ breaks (ii). Only (C) $4 < x < 5$ matches.
Spot check with $x = 4.5$ : $2 (4.5) - 1 = 8 > 7$ and $2 (4.5) - 9 = 0 < 1$ . Both hold.

KEY IDEA

Solve each linear inequality on its own, then intersect the two solution sets. "Satisfies both" means the overlap of the sets, not their union. Keep each end strict or non-strict exactly as the original inequality was.

WATCH OUT FOR

Picking (B) $- 4 < x \leq 4$ , whether from misreading $2 x - 1 > 7$ as $x > - 4$ or from treating the endpoint 4 as included. At $x = 4$ , $2 (4) - 1 = 7$ , which is not greater than 7, so 4 is excluded. Another slip is taking the union of the two sets, which is every real number, instead of their intersection.

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GATE 2022 EC GA Q.7: solution and answer

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