Q.State whether the following statement is True or False: If $x < y$ and $b < 0$, then $\dfrac{x}{b} < \dfrac{y}{b}$.

Concept: Linear Inequality Solutions — Multiplying or dividing an inequality by a negative number reverses the inequality sign.

Step 1: We are given $x < y$ and $b < 0$ (so $b$ is negative).

Step 2: Dividing both sides of $x < y$ by $b$ (a negative number) requires flipping the inequality sign:

$\frac{x}{b} > \frac{y}{b}.$ …

The core idea here is how inequalities behave under multiplication or division by negative numbers. Many students learn the rule "flip the sign when multiplying or dividing by a negative" but sometimes forget to apply it in the heat of the moment. Let's break down exactly why this rule exists and why the given statement fails.

Think of an inequality like $x < y$ as saying "$x$ is to the left of $y$ on the number line." When you divide both sides by a negative number, you're essentially reflecting the number line about zero — left becomes right and right becomes left. That reflection is why the inequality direction must reverse.

Now, let's work through the problem step by step.

1. Start with the given inequality:

   We have $x < y$. This means $x$ is strictly less than $y$.

2. Consider the condition on $b$:

   We are told $b < 0$, so $b$ is a negative number. For example, $b$ could be $-1$, $-5$, or any negative real number.

3. Divide both sides of $x < y$ by $b$:

   Since $b$ is negative, dividing by $b$ is equivalent to multiplying by $\frac{1}{b}$, which is also negative (the reciprocal of a negative number is negative).

   The rule for inequalities is:

   If $a < b$ and $c < 0$, then $a \cdot c > b \cdot c$ (and similarly for division).

   Multiplying or dividing an inequality by a negative number reverses the inequality sign.

4. Apply the rule:

   Dividing $x < y$ by the negative number $b$ gives:

5. Compare with the statement: …