Algebra · Step-by-step guide
How to Solve Inequalities
Solve and graph linear inequalities, including the rule for multiplying or dividing by a negative.
Start with the central idea
Solve an inequality like an equation, but reverse the inequality symbol whenever you multiply or divide both sides by a negative number. Graph < or > with an open point and ≤ or ≥ with a closed point.
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Preserve order
Adding or subtracting the same amount keeps the inequality direction. Multiplying or dividing by a negative reverses the order.
Show the boundary
An open point excludes the endpoint; a closed point includes it.
Test when unsure
Choose a value from the shaded region and substitute it into the original inequality.
See the structure
A picture makes the relationships easier to remember.
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
Solve .
- Add 5: .
- Divide by 3: .
- Use an open point at 5 and shade left.
Solve .
- Subtract 1: .
- Divide by −4 and reverse the symbol.
- Check x = −3 gives equality.
Solve .
- Subtract 5 from all three parts.
- Keep both comparison symbols.
- Graph an open point at −3 and a closed point at 4.
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Reversing the symbol after adding a negative rather than multiplying or dividing by one.
Using the wrong endpoint circle.
Applying an operation to only two parts of a compound inequality.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Solving Inequalities
Practice one-step, multi-step, and compound inequalities.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Solve .
Show hint
Isolate x.
Show worked solution
- Add 7: .
- Divide by 5.
- The sign stays the same.
Answer:
Solve .
Show hint
Reverse the sign when dividing by −3.
Show worked solution
- Subtract 2: .
- Divide by −3 and reverse ≥.
- .
Answer:
Solve .
Show hint
Distribute first.
Show worked solution
- .
- .
- .
Answer:
Solve .
Show hint
Apply each operation to all three parts.
Show worked solution
- Subtract 1: .
- Divide all parts by 3.
- .
Answer:
Solve .
Show hint
Subtract 2x from both sides.
Show worked solution
- The variable terms cancel.
- The result is , which is false.
- No value of x works.
Answer: No solution.
Frequently asked questions
Quick answers before you move on.
What should I remember first?
Solve an inequality like an equation, but reverse the inequality symbol whenever you multiply or divide both sides by a negative number. Graph < or > with an open point and ≤ or ≥ with a closed point.
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.