Dividing both sides by −4 requires flipping the inequality sign: ≥ becomes ≤. So −4x ≥ 20 becomes x ≤ −5. Verify: try x = −6 (satisfies x ≤ −5): −4(−6) = 24 ≥ 20 ✓. Try x = −4 (does not satisfy x ≤ −5): −4(−4) = 16 ≥ 20? No ✗. The most common error is forgetting to flip the sign, producing the wrong answer x ≥ −5.
Question 2 Multiple Choice
Which number line graph correctly represents the solution to x + 5 < 9?
AClosed circle at 4, shaded to the right
BOpen circle at 4, shaded to the right
COpen circle at 4, shaded to the left
DClosed circle at 14, shaded to the left
Solving: x + 5 < 9 → x < 4. The solution is all numbers less than 4, which means shading to the left of 4. The inequality is strict (< not ≤), so 4 itself is NOT included — use an open circle. A closed circle would incorrectly include 4; shading to the right would represent x > 4, the opposite of the correct solution.
Question 3 True / False
When solving an inequality, multiplying both sides by a negative number does not change the direction of the inequality sign.
TTrue
FFalse
Answer: False
Multiplying or dividing both sides by a negative number always reverses the inequality direction. The reason: multiplying by −1 reflects all numbers across zero on the number line, reversing their order. For example, 2 < 5, but multiplying both by −1 gives −2 > −5. Since inequalities describe left-right order on the number line, this reflection flips every 'greater than' to 'less than' and vice versa. This rule has no counterpart in equations and is the single most common error when solving inequalities.
Question 4 True / False
The solution to an inequality like x − 3 > 7 is a set of infinitely many numbers, not a single value.
TTrue
FFalse
Answer: True
Solving x − 3 > 7 gives x > 10. Every number greater than 10 satisfies this — 10.001, 11, 100, 1,000,000, and infinitely many others. This is a fundamental difference from equations: x − 3 = 7 has exactly one solution (x = 10), while x − 3 > 7 describes an infinite range. The solution is represented as a ray on the number line (starting at an open circle at 10, shading right), not a single point.
Question 5 Short Answer
Explain why the inequality sign must reverse when you multiply or divide both sides by a negative number. Use the number line to justify your answer.
Think about your answer, then reveal below.
Model answer: On the number line, multiplying every number by −1 is a reflection across zero: positives become negatives and vice versa. This reflection reverses the ordering of all numbers — what was to the right is now to the left. For example, 3 is to the right of 1 (3 > 1), but after multiplying by −1, −3 is to the left of −1 (−3 < −1). Since inequalities describe this left-right ordering, the reflection flips every 'greater than' into 'less than' and vice versa. Failing to flip produces an inequality satisfied by all the wrong values.
A concrete check reinforces this: start with 2 < 10, multiply both sides by −2: you get −4 and −20. Is −4 < −20? No — −4 > −20. The sign had to flip. Substitution checks like this are the best safeguard against sign-flip errors when solving inequalities.