Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for 7th grade two-step inequalities? Then this Grade 7 lesson on Solving and Graphing Two-Step Inequalities explains each idea step by step, with worked examples.
Quick answer: A two-step inequality, such as 4x − 5 > 11, is solved like an equation: undo addition or subtraction, then undo multiplication or division. Reverse the inequality sign whenever you multiply or divide both sides by a negative number. Graph the solution set with an open circle and a shaded ray, then interpret it in context.
Standards: 7.EE.B.4b · Level: Grade 7
What is the solution set of an inequality?
An inequality such as 4x − 5 > 11 has many solutions, not just one. Its solution set contains every number that makes the statement true, and a graph on a number line shows the whole set at once.
Use an open circle at the endpoint for > or <, because the endpoint itself is not a solution. Shade toward the numbers that make the inequality true.
How do you solve px + q > r?
Use the same inverse-operation steps as for equations: undo the addition or subtraction first, then undo the multiplication or division.
Solve 4x − 5 > 11.
4x > 16 so x > 4
Test 5: 4(5) − 5 = 15, which is greater than 11. The test passes.
Why does the inequality sign reverse for a negative number?
Multiplying by a negative moves numbers to the opposite side of zero, which reverses their order. Start with the true statement 2 < 5.
| Operation on 2 < 5 | Result | True statement |
|---|---|---|
| Multiply by 3 | 6 and 15 | 6 < 15 |
| Multiply by −3 | −6 and −15 | −6 > −15 |
When you multiply or divide both sides by a negative number, reverse the inequality sign. Adding or subtracting never changes it.
- Undo the addition or subtraction.
- Undo the multiplication or division, reversing the sign if the number is negative.
- Graph with an open circle and shade the correct side.
- Test one number from the shaded part in the original inequality.
For example, −3x + 4 < 19 gives −3x < 15, and dividing by −3 gives x > −5.
How do you interpret a solution in context?
A streaming plan costs $20 plus $1.50 per movie rental, and the budget is under $32. The inequality 20 + 1.5t < 32 gives t < 8, so any number of rentals below 8 stays within budget.
Subtracting a negative number or having a negative constant does not reverse the sign. Only multiplying or dividing by a negative does.
Always test one point from your shaded ray. If it fails the original inequality, the sign or the direction is wrong.
- An inequality usually has infinitely many solutions, shown as a ray on a number line.
- Use an open circle for the symbols > and <, since the endpoint is not included.
- Multiplying or dividing both sides by a negative number reverses the inequality sign.
- Testing one number from the shaded ray in the original inequality catches most mistakes.
Solving and Graphing Two-Step Inequalities: frequently asked questions
When do I reverse the inequality sign?
Reverse it only when you multiply or divide both sides by a negative number. Adding or subtracting any number, including a negative one, leaves the sign unchanged.
What does an open circle mean on the graph?
It marks an endpoint that is not part of the solution set. For x > 3, the number 3 itself does not make the inequality true, so the circle stays open.
How can I check a solution of an inequality?
Pick a number from the shaded region, substitute it into the original inequality, and confirm the statement is true. Also try the endpoint, which should fail.
Why can a real-life answer differ from the graph?
Contexts such as tickets or shirts require whole numbers. The graph includes decimals, so an answer like n > 22.5 becomes “at least 23” in a story.
Is x < 5 the same as 5 > x?
Yes. Both statements say that x is less than 5. Reading the inequality with the variable first makes the graph easier to draw.
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