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When working with inequalities, multiplying both sides of an inequality by a number helps to keep the relationships consistent, provided you remember this key rule: the sign does not change if the number is positive.
For example, consider the inequality: \(3 < 7\). If you multiply both sides by a positive number, like 4, the inequality becomes \(4 \times 3 < 7 \times 4\), which gives \(12 < 28\). Notice how the inequality symbol stays the same.
This is because you're increasing both sides of the inequality equally, so their relationship does not change.

Just like multiplication, division can also be applied to both sides of an inequality without altering the inequality sign, but there's an important distinction to keep in mind. If you divide by a positive number, the inequality sign remains the same.
However, if dividing by a negative number, the inequality sign must flip. Here's an example: \(24d < 32\). Dividing by a positive 3 maintains the inequality sign: \(\frac{24d}{3} < \frac{32}{3}\), resulting in \(8d < \frac{32}{3}\). But if we divide both sides of the inequality \(24x \leq 32\) by -3, we flip the inequality sign: \(\frac{24x}{-3} \geq \frac{32}{-3}\), which converts to \(-8x \geq -\frac{32}{3}\). Remember, division by negatives flips the direction!

Adding or subtracting the same number on both sides of an inequality does not alter the direction of the inequality. This is because both sides are being adjusted by the same amount, maintaining the original order or relationship. For instance, consider the inequality: \(-4 \geq x\). Adding -10 to both sides can be seen as subtracting 10. The inequality becomes \(-4 + (-10) \geq x + (-10)\), simplifying to \(-14 \geq x - 10\), where the inequality sign remains unchanged. This concept is essential for balancing equations and keeping inequalities accurate.

Understanding how negative and positive numbers affect inequalities is crucial. The direction of the inequality may remain unchanged or need flipping based on the operations involved and the nature of the numbers. - **Positive Multiplication/Division**: Maintain the inequality direction. \((e.g.,\ 3 < 7,\ 12 < 28)\)- **Negative Multiplication/Division**: Flip the inequality direction. \((e.g.,\ 5 \leq 12, -15 \geq -36)\)- **Adding/Subtracting Any Number**: The inequality's direction does not change. \((e.g.,\ -4 \geq x,\ -14 \geq x - 10)\)Remember these principles to solve inequalities confidently.