If \(\displaystyle\frac{2-x}{4}-\frac{4+x}{6}\ge 10\) then :
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(x\le -\displaystyle\frac{122}{5}\)
Multiplying both sides by 12 (positive, so the sign stays the same):
\(3(2 - x) - 2(4 + x) \ge 120\)
\(6 - 3x - 8 - 2x \ge 120\)
\(-5x - 2 \ge 120\)
\(-5x \ge 122\)
Dividing by −5, a negative number, reverses the inequality:
\( \displaystyle x \le -\frac{122}{5} \)
Why the other options are incorrect
Option (1) \(x\ge -\displaystyle\frac{122}{5}\) — Incorrect. This keeps the sign \(\ge\) after dividing by −5; the sign must reverse.
Option (2) \(x\ge -\displaystyle\frac{5}{122}\) — Incorrect. The fraction is upside down and the sign is not reversed.
Option (4) \(x\le -\displaystyle\frac{5}{122}\) — Incorrect. The fraction is upside down: x is \(-\frac{122}{5}\) or less, not \(-\frac{5}{122}\) or less.