Inequalities
Linear inequalities, representing solutions on number lines, and (Higher) quadratic and graphical inequalities.
Watch the lesson
Every point below, worked through in one video, and each point links straight to its own part.
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Inequalities, explained point by point
Everything the GCSE specification expects you to be able to do, and how to actually do it - the same lesson a signed-in student studies from.
Use inequality notation and show inequalities on a number line
x > 2 means anything bigger than 2; x ≥ 2 includes 2 itself. On a number line, an open circle means the end value is NOT included (< or >) and a filled circle means it is (≤ or ≥), with an arrow or bar showing the covered region.
Solve linear inequalities in one variable
Solve exactly like an equation, keeping the inequality sign: 3x + 2 < 11 gives x < 3. The one exception: multiplying or dividing by a NEGATIVE number flips the sign - from −2x < 6 you get x > −3. Test a value in the original to check.
List the integer solutions of an inequality
Solve first, then read off the whole numbers. For −3 < x ≤ 2 the integers are −2, −1, 0, 1, 2 - the strict < excludes −3, the ≤ includes 2. Double-inequalities like −5 ≤ 2x + 1 < 7 are solved on all three parts at once.
Solve quadratic inequalitiesHigher only
Solve the matching equation first, then sketch the parabola to see where it is above or below zero. x² − x − 6 < 0 has roots −2 and 3; the U-shape dips below the axis BETWEEN them, so −2 < x < 3. "Greater than" picks the two outside regions instead.
Show inequalities as regions on a graphHigher only
Draw each boundary line (solid for ≤/≥, dashed for </>) and shade according to the inequality - test a point like (0, 0) if unsure which side. With several inequalities, the answer is the region satisfying all of them at once; label it R.
Common mistakes in Inequalities
The slips that cost marks year after year, and what to do instead.
Keeping the same sign after dividing by a negative, so −2x < 6 becomes x < −3.
Multiplying or dividing by a negative flips the sign: −2x < 6 gives x > −3. Test x = 0, which fits −2x < 6 and x > −3 but not x < −3.
Using a filled circle for < or >, or an open circle for ≤ or ≥, on a number line.
An open circle means the end value is not included, and a filled circle means it is. So x > 2 has an open circle at 2.
Including the end of a strict inequality when listing integers, so the list for −3 < x ≤ 2 starts at −3.
The < leaves −3 out and the ≤ lets 2 in, so the integers are −2, −1, 0, 1 and 2.
Swapping the inequality for an equals sign and finishing with x = 3 instead of x < 3.
Keep the inequality sign on every line so the answer is a range: 3x + 2 < 11 gives 3x < 9, so x < 3.
Solving x² < 9 as just x < 3 and losing the negative side.Higher only
Solve x² = 9 to get −3 and 3, then sketch y = x² − 9: it is below zero between the roots, so −3 < x < 3.
Practice
Try a Inequalities question
A GCSE-style original question from this topic. Have a go before you open the working - deciding on an answer first is what makes the working stick.
Answer: x > 5
Collect the x terms on one side: 2x > 10, so x > 5.
- Subtract 2x from both sides: 2x − 3 > 7.
- Add 3 to both sides: 2x > 10.
- Divide both sides by 2: x > 5.
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