Lesson 7 of 1524
Inequalities
Keep both sides in step, and flip the sign when you multiply or divide by a negative.
Practice this chapter< means less than and > means greater than. The line under ≤ or ≥ adds “or equal.”
You may add or subtract the same number on both sides without flipping. Multiplying or dividing by a negative number flips the inequality.
A statistics test uses this language directly: reject the null when the p-value is less than or equal to the cutoff.
Dividing by a negative
If −2x < 6, then x > −3
Dividing by −2 reverses the direction.
Worked example
Solve 2x + 1 < 9.
- 1Subtract 1 from both sides. Addition and subtraction do not flip the inequality: 2x < 8.
- 2Divide both sides by 2. The divisor is positive, so the sign keeps its direction: x < 4.
- 3Check a value that should work. If x = 0, then 2(0) + 1 = 1, and 1 is less than 9.
- 4Check a value that should fail. If x = 5, then 2(5) + 1 = 11, and 11 is not less than 9.
Result: x < 4
Why. The inequality stays pointed the same way because you divided by positive 2. It would have flipped only if that divisor had been negative. The checks show that numbers below 4 work and numbers above 4 do not.
Forgetting to flip the sign when you divide by a negative turns the answer around.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.