Guides / Inequalities

Flipping the inequality sign: when and why

The rule, the reason behind it, and the places students flip when they shouldn't (or don't when they should).

Why multiplying by a negative flips it

Start with something true: −3 < 2. Multiply both sides by −1 and you get 3 and −2. But 3 < −2 is false. The true statement is 3 > −2.

Multiplying by −1 reflects every number across zero on the number line. What was to the left ends up on the right, so the order reverses. Multiplying by any other negative number does the same reflection plus a stretch, so the order reverses there too. Adding or subtracting just slides both numbers the same distance, and multiplying by a positive number stretches them, and neither of those changes which one is on the left.

Example 1: −2x + 5 > 11

1. −2x + 5 > 11Start.
2. −2x > 6Subtract 5 from both sides. No flip: subtracting never flips.
3. x < −3Divide both sides by −2. Flip, because −2 is negative.

Test one number on each side of −3 in the original inequality. x = −4: 8 + 5 = 13 > 11, true. x = 0: 5 > 11, false. The solution really is the numbers below −3.

Example 2: you can often avoid the flip

Solve 4 − x ≤ 3x + 12 two ways.

1. 4 − x ≤ 3x + 12Start.
2. −4x ≤ 8Subtract 3x and subtract 4 from both sides.
3. x ≥ −2Divide by −4 and flip.

Or move the x terms to the side where they'll be positive:

1. 4 − x ≤ 3x + 12Start.
2. −8 ≤ 4xAdd x and subtract 12 from both sides.
3. −2 ≤ xDivide by 4. Positive, so no flip.

Same answer, since −2 ≤ x and x ≥ −2 say the same thing. If negatives make you nervous, collecting x where its coefficient is positive means you never have to divide by a negative.

Example 3: a negative fraction

1. −x3 ≥ 2Start.
2. x ≤ −6Multiply both sides by −3 and flip.

Example 4: a compound inequality

Solve −3 < 1 − 2x ≤ 7. Do each operation to all three parts, and flip both signs when you divide by the negative.

1. −3 < 1 − 2x ≤ 7Start.
2. −4 < −2x ≤ 6Subtract 1 from all three parts.
3. 2 > x ≥ −3Divide all three by −2 and flip both signs.
4. −3 ≤ x < 2Rewrite smallest to largest. Same statement, easier to read.

Check the endpoints: at x = −3, 1 − 2x = 7, which is allowed by ≤ 7, so −3 is included. At x = 2, 1 − 2x = −3, which fails −3 <, so 2 is excluded.

Common mistakes, and the exact line they happen on

1. Flipping because a negative number is involved

The rule is about what you multiply or divide by, not about whether negative numbers appear anywhere.

1. 3x > −12Start.
2. x < −4This is the mistake. Dividing by 3, which is positive. The −12 doesn't matter. It should stay x > −4.

1. x − 5 < −2Start.
2. x > 3This is the mistake. Adding 5 never flips. It should be x < 3.

2. Forgetting the flip

1. −2x > 6Fine so far.
2. x > −3This is the mistake. Divided by −2 without flipping. Test x = 0: −2(0) > 6 is false, but 0 > −3 is true, so the two lines don't agree.

A quick test value catches both mistakes. Pick an easy number your answer says should work, and plug it into the original inequality.

3. Multiplying by a variable

If you multiply both sides by x, you don't know whether to flip, because you don't know whether x is positive or negative. Avoid it, or split the problem into the x > 0 and x < 0 cases.

Practice

1. Solve −5x ≥ 20.

   Show answer

   Divide by −5 and flip: x ≤ −4.

2. Solve 3x − 7 < 8.

   Show answer

   3x < 15, so x < 5. No flip anywhere.

3. Solve 2 − 3x > 14.

   Show answer

   −3x > 12, then divide by −3 and flip: x < −4.

4. Solve −x/2 + 1 ≤ 4.

   Show answer

   −x/2 ≤ 3, then multiply by −2 and flip: x ≥ −6.

5. Solve 4x > −8.

   Show answer

   x > −2. Dividing by positive 4, so no flip, even though −8 is negative.

6. Solve −1 ≤ 3 − 2x < 5.

   Show answer

   Subtract 3: −4 ≤ −2x < 2. Divide by −2 and flip both: 2 ≥ x > −1, which is −1 < x ≤ 2.