A student solves −2x > 8 and gets x > −4. What error did the student make?
AThe student moved the constant to the wrong side
BThe student divided both sides by −2 but forgot to flip the inequality sign, so the correct answer is x < −4
CThe student made an arithmetic error; the correct answer is x > −16
DNo error — x > −4 is the correct solution
Dividing both sides by −2 requires flipping the inequality: −2x > 8 becomes x < −4 (not x > −4). The sign reversal happens because dividing by a negative flips the number line ordering — what was greater becomes lesser. The student correctly performed the division but forgot the sign flip, which is the most common error in inequality solving.
Question 2 Multiple Choice
Solve −3x + 7 > 1. Which of the following is the correct solution?
Ax > 2
Bx < 2
Cx > −2
Dx < −2
Subtract 7 from both sides: −3x > −6. Divide both sides by −3 — and flip the sign: x < 2. The flip occurs because we divided by a negative number. Verify: x = 0 gives −3(0) + 7 = 7 > 1 ✓ (0 is in the solution set x < 2). x = 3 gives −3(3) + 7 = −2, and −2 > 1 is false ✓ (3 is correctly excluded).
Question 3 True / False
The solution to a linear inequality with one variable is always a range of values (an interval on the number line), never a single number.
TTrue
FFalse
Answer: True
An inequality like x < 2 describes infinitely many values — all real numbers less than 2. Unlike an equation (which has a single solution point), an inequality defines a region. Graphically, this is represented with shading extending in one direction from a boundary value. Thinking the answer 'should be a number' is a common error that comes from confusing inequalities with equations.
Question 4 True / False
You should flip the inequality sign whenever you subtract a positive number from both sides of an inequality.
TTrue
FFalse
Answer: False
The sign only flips when you multiply or divide both sides by a negative number. Adding or subtracting any number (positive or negative) from both sides preserves the direction of the inequality. For example, x + 3 > 7 → x > 4 (subtracting 3, no flip). Only multiplication and division by negatives cause the flip, because they reverse the ordering relationship on the number line.
Question 5 Short Answer
Explain why multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Use a numeric example to illustrate.
Think about your answer, then reveal below.
Model answer: Multiplying by a negative flips the number line — larger numbers become smaller and vice versa. Example: 3 > 1 is true. Multiply both sides by −1: −3 and −1. Now −3 < −1, so the relationship reverses. The same logic applies when solving: dividing by −3 flips every comparison.
The flip is not an arbitrary rule — it follows directly from how negative numbers reorder the number line. Positive multiplication preserves order (if a > b and c > 0, then ac > bc). Negative multiplication reverses it (if a > b and c < 0, then ac < bc). Understanding the reason makes the rule memorable and prevents forgetting it under exam pressure.