Inequalities Worksheets

Solve and write the set

One-step problems undo a single operation: x + 4 > 9, or 3x ≤ 12, or x/2 < 5. Two-step problems need two: 3x − 4 ≥ 5, or −2x + 7 < 1. Mixed blends them.

Sixteen questions by default, answer key optional. The answer wanted is a solution set — "x < 5" — not a single number.

Almost the same as an equation

Which is the good news and the trap.

Everything you do to solve 3x − 4 = 5 works identically on 3x − 4 ≥ 5. Add to both sides, subtract from both sides, divide both sides. The method transfers wholesale, so a class that can solve equations can solve most of these immediately.

Then there is the flip

Multiply or divide both sides by a negative number and the inequality sign reverses.

This is the one genuinely new rule, and it is not arbitrary. Start with 2 < 5, which is true. Multiply both sides by −1 and you have −2 and −5 — and −2 > −5, because further left is smaller. The direction had to turn over for the statement to stay true.

Show that once with numbers before stating the rule. A class that has seen it happen remembers; a class that has been told flips inconsistently for a year.

Two-step mode is where it bites

The one-step problems rarely need the flip. The two-step set includes negative coefficients precisely so it comes up, and −2x + 7 < 1 is the shape to expect trouble on.

An alternative worth teaching to anyone who keeps forgetting: move the x term to whichever side keeps it positive. −2x + 7 < 1 becomes 6 < 2x, and no flip is ever required. It is slightly longer and it removes the error entirely.

Checking an answer

Pick any number in your solution set and put it back in the original.

If x < 5, try 0. If the original statement is true, the set is probably right; if it is false, either the arithmetic or the direction has gone wrong. Then try a number outside the set and confirm it fails. Thirty seconds, and it catches every flip error there is.

Notation

Keep the variable on the left when writing the answer. "5 > x" and "x < 5" say the same thing and one of them is much easier to read at a glance, especially when a pupil is later asked to sketch it on a number line.

Level: roughly thirteen to sixteen. Related: equation solving, which should come first, and negative numbers.