Inequalities and their solution sets
Understand inequalities as statements comparing expressions, represent solutions on a number line, and solve simple linear inequalities using the same inverse-operation methods as equations
What a learner can do afterwards
- Write an inequality from a worded constraint (e.g., 'must be at least 12' → x ≥ 12)
- Represent the solution set of an inequality on a number line with open or closed circles
- Solve a one-step or two-step inequality such as 3x + 1 < 10
1 · Read
You already know how to solve an equation like x + 3 = 10, where both sides end up exactly equal. An inequality compares two expressions without an equal sign. It uses one of four symbols: < (less than), > (greater than), ≤ (less than or equal to, or 'at most'), and ≥ (greater than or equal to, or 'at least').
A club rule says, 'You must be at least 12 to join.' If x stands for your age, that rule becomes x ≥ 12. The words 'at least' turn into ≥, because 12 works and so does any age above it.
The solutions of an inequality are a whole set of numbers, so a number line shows them as a shaded stretch instead of one point. The boundary is the number in the inequality. Fill the circle for ≤ or ≥, because the boundary is itself a solution. Leave the circle open for < or >, because the boundary is not. Then shade the side that makes the statement true.
Solve 3x + 1 < 10 the same way you'd solve an equation: undo the operations in reverse order. Subtract 1 from both sides: 3x < 9. Divide both sides by 3: x < 3. Every number smaller than 3 makes the inequality true.
Check your answer by putting a number from your solution set back into the original inequality. If the statement comes out true, your boundary and your direction are both right.
An inequality compares expressions with <, >, ≤, or ≥; solve it the same way you solve equations, then show the answer as a range on a number line with an open or closed circle.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.