Explain how solving -7y > 161 is different from solving 7y > -161.
The correct answer and explanation is:
Correct Answer:
Solving -7y > 161 is different from solving 7y > -161 because when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign. This is not required when dividing or multiplying by a positive number.
Detailed Explanation (300 words):
Inequalities, like equations, are solved using inverse operations. However, they include an important rule:
When you multiply or divide both sides of an inequality by a negative number, you must flip (reverse) the inequality sign.
Let’s solve both inequalities to see the difference.
1. Solving -7y > 161
- We want to isolate y, so we divide both sides by -7.
- Since we are dividing by a negative, we flip the inequality sign:
−7y−7<161−7\frac{-7y}{-7} < \frac{161}{-7} y<−23y < -23
Answer: y<−23y < -23
2. Solving 7y > -161
- We divide both sides by 7, which is a positive number. So, we keep the inequality sign:
7y7>−1617\frac{7y}{7} > \frac{-161}{7} y>−23y > -23
Answer: y>−23y > -23
Key Differences:
- In -7y > 161, dividing by -7 flips the inequality to <.
- In 7y > -161, dividing by 7 keeps the inequality as >.
This change in the direction of the inequality is essential because it reflects the reversal of order that happens when you scale both sides by a negative number. Failing to flip the sign leads to incorrect solutions that don’t accurately represent the values that satisfy the original inequality. Understanding this rule is critical in algebra to preserve the correct relationship between the expressions.