Q.The solution of the inequation $-2x>6$ over $\mathbb{R}$ is: (A) $x>-3$ (B) $x<-3$ (C) $x>3$ (D) $x<3$

Divide by the negative coefficient and reverse the sign.

$-2x>6\Rightarrow x<\dfrac{6}{-2}=-3$ (sign reversed on dividing by $-2$).

Divide both sides of $-2x>6$ by $-2$. Since $-2$ is negative, the inequality sign reverses from $>$ to $<$:

So the solution is $x<-3$, i.e. $(-\infty,-3)$.

Why the other options are wrong:

• (A) $x>-3$ and (C) $x>3$ keep the sign as $>$, the classic error of not flipping when dividing by a negative.
• (D) $x<3$ divides by $2$ instead of $-2$, losing the negative sign on the boundary. Only (B) both flips the sign and uses the correct boundary $-3$.

Check (independent verification): test $x=-4$ (should satisfy (B)): $-2(-4)=8>6$ ✓. Test $x=0$ (should fail): $-2(0)=0>6$ is false ✓.

Failing to reverse the sign when dividing by $-2$ — the single most common slip, and exactly what options (A) and (C) test.