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.
13 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.