Q.Given that $x$, $y$ and $b$ are real numbers and $x < y$, $b < 0$, then
(A) $\dfrac{x}{b} < \dfrac{y}{b}$
(B) $\dfrac{x}{b} \le \dfrac{y}{b}$
(C) $\dfrac{x}{b} > \dfrac{y}{b}$
(D) $\dfrac{x}{b} \ge \dfrac{y}{b}$

The key idea here is understanding how operations with negative numbers affect the direction of an inequality.

1. We are given the inequality $x < y$.
2. We need to divide both sides of this inequality by $b$.
3. A crucial piece of information is that $b < 0$, meaning $b$ is a negative real number. …

Understanding how inequalities behave when you perform operations on them is fundamental. The key concept here revolves around what happens when you multiply or divide an inequality by a negative number.

Think about a simple number line. If you have two numbers, say $2$ and $5$, we know that $2 < 5$.

Now, let's multiply both by a positive number, for example, $3$:

$2 \times 3 = 6$

$5 \times 3 = 15$

The inequality $6 < 15$ still holds true. The direction of the inequality sign remains the same.

However, consider what happens when you multiply or divide by a negative number. Let's take our original $2 < 5$ and multiply both sides by $-1$:

$2 \times (-1) = -2$

$5 \times (-1) = -5$

Now, compare $-2$ and $-5$. On the number line, $-2$ is to the right of $-5$, which means $-2$ is greater than $-5$.

So, $-2 > -5$.

Notice that the original inequality sign ($<$) has flipped to ($>$). This is a crucial rule for inequalities.

This rule applies because multiplying or dividing by a negative number essentially "reflects" the numbers across zero on the number line, reversing their relative order.

Let's apply this to the given problem.

1. Start with the given inequality:

   We are given that $x$ and $y$ are real numbers such that $x < y$.

2. Identify the operation and the condition:

   We need to compare $\dfrac{x}{b}$ and $\dfrac{y}{b}$. This means we are dividing both sides of the inequality $x < y$ by $b$.

   We are also given a critical condition: $b < 0$. This tells us that $b$ is a negative number.

3. Apply the rule for dividing by a negative number: …