In this guide
What you will be able to do
- Work each problem without looking at the answer.
- Write the algebraic step that justifies each change.
- Check the result in the original relationship.
- Use the error notes to decide what to practise next.
How to use this worksheet
Solve each inequality, reverse the symbol only when multiplying or dividing by a negative number, and show the result on a number line when requested.
- Use inverse operations
- Reverse inequality after division by a negative
- Solve compound inequalities
- Interpret bounds in context
For a clean student copy, print this page and cover the worked solutions or select the problem pages in your browser’s print preview.
Practice problems
Problem 1
3x + 5 < 20
Problem 2
−4x ≥ 12
Problem 3
2(x − 3) + 1 > 9
Problem 4
−2 ≤ 3x + 4 < 13
Problem 5
A taxi budget is at most $40. The base fee is $7 and each mile is $2.75. How many miles m can be travelled?
Answer key with worked solutions
Problem 1
3x + 5 < 20
- 1.Subtract 5: 3x < 15.
- 2.Divide by 3: x < 5.
Answer: x < 5
Check: x = 4 works; x = 5 does not.
Problem 2
−4x ≥ 12
- 1.Divide by −4 and reverse the symbol.
Answer: x ≤ −3
Check: x = −3 gives 12 ≥ 12.
Problem 3
2(x − 3) + 1 > 9
- 1.Distribute: 2x − 6 + 1 > 9.
- 2.2x − 5 > 9, so 2x > 14.
- 3.x > 7.
Answer: x > 7
Check: x = 8 gives 11 > 9.
Problem 4
−2 ≤ 3x + 4 < 13
- 1.Subtract 4 throughout: −6 ≤ 3x < 9.
- 2.Divide throughout by 3.
Answer: −2 ≤ x < 3
Check: Both boundary conditions match the original.
Problem 5
A taxi budget is at most $40. The base fee is $7 and each mile is $2.75. How many miles m can be travelled?
- 1.Write 7 + 2.75m ≤ 40.
- 2.Subtract 7: 2.75m ≤ 33.
- 3.Divide: m ≤ 12.
Answer: At most 12 miles
Check: 7 + 2.75(12) = 40.
Use mistakes to choose the next step
Review linear equationsUse the beginner guide
| If this happened | Likely issue | Next practice move |
|---|---|---|
| Symbol reversed after addition | Rule overgeneralised | Reverse only for multiplication or division by a negative |
| Open/closed endpoint wrong | Boundary inclusion confusion | ≤ or ≥ includes the boundary |
| Context answer ignores whole units | Interpretation issue | Round only in the direction that keeps the inequality true |
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Useful answers
Frequently asked questions
Why does the inequality sign reverse with a negative?
Multiplying or dividing by a negative reverses order on the number line, so the relationship direction must reverse.
What does an open circle mean?
The boundary value is excluded, as with < or >. A closed circle includes it, as with ≤ or ≥.
Sources, scope and accuracy
The worked examples and practice questions on this page are original Cambridge Veritas learning materials, not released STAAR items. Curriculum and testing details can change; use the linked Texas Education Agency pages for current official information. Cambridge Veritas is not affiliated with or endorsed by TEA.