Question

If $a>b$ and $c<0$, then which of the following is true?

• A. $a + c<b + c$
• B. $a - c<b - c$
• C. $a c>b c$
• D. $a - c>b + c$

Hint:

When multiplying or comparing inequalities with negative numbers, be careful—the direction of the inequality may flip.

Answer 1

The Correct Option is C

Given $a>b$ and $c<0$. Multiplying $a>b$ by a negative number $c$ reverses the inequality. Thus, $a.c < b.c$. Option (C) states $a c>b c$, which is incorrect. Rechecking: since $c<0$ and $a>b$, multiplying by $c$ yields $a c<b c$. The correct relation is $a c<b c$, not $a c>b c$. Hence, option (C) is incorrect.
Testing each option with numbers:
Let $a = 5$, $b = 3$, $c = -2$.
(A): $5 + (-2) = 3$, $3 + (-2) = 1$. $3<1$ is false.
(B): $5 - (-2) = 7$, $3 - (-2) = 5$. $7<5$ is false.
(C): $5 \times (-2) = -10$, $3 \times (-2) = -6$. $-10>-6$ is false.
(D): $5 - (-2) = 7$, $3 + (-2) = 1$. $7>1$ is true.
If $a>b$ and $c<0$, then $-c$ is positive. Thus, $a - c$ increases $a$, and $b + c$ decreases $b$. Therefore, $a - c>b + c$ is a true statement.