Algebra · Grade 8 · Study notes
Linear Equations and Inequalities
An equation says two expressions are equal. An inequality compares values using signs such as <, >, ≤ and ≥.
Learning time: 20–25 minutes · Level: Developing
Learning goals
- Solve multi-step linear equations using inverse operations and the distributive property.
- Collect variable terms on one side and recognise one-solution, no-solution and infinitely-many-solution cases.
- Solve and graph linear inequalities, including reversing the sign when multiplying or dividing by a negative number.
- Check equation solutions by substitution and test values in inequality solutions.
1. Keep equations balanced
An equation stays equivalent when you perform the same operation on both sides. Use inverse operations to isolate the variable.
2. Multi-step equations
Simplify each side first, use the distributive property when needed, collect variable terms on one side, then isolate the variable. Always substitute your answer to check.
3. Variables on both sides
For 5x + 2 = 2x + 14, subtract 2x from both sides, then subtract 2: 3x = 12, so x = 4. If the variable terms cancel, the result may be always true (infinitely many solutions) or false (no solution).
4. Inequalities
Solve an inequality like an equation, but reverse the inequality sign when multiplying or dividing both sides by a negative number. For example, −2x < 8 becomes x > −4.
5. Show solutions
Use an open circle for < or > and a filled circle for ≤ or ≥ on a number line. Shade in the direction of values that satisfy the inequality. Test a value from the shaded region.
Choose the right method and check the result
For an equation with variables on both sides, collect variable terms on one side and constants on the other. Keep the equation balanced by performing the same operation on both sides. For an inequality, use the same process, but reverse the inequality sign whenever you multiply or divide by a negative number.
Worked reasoning: Solve 5x − 7 = 2x + 8. Subtract 2x from both sides to get 3x − 7 = 8. Add 7 to both sides: 3x = 15. Divide by 3: x = 5. Substitution gives 25 − 7 = 2(5) + 8, so both sides equal 18.
For −2x < 8, divide by −2 and reverse the sign: x > −4. A quick test with x = 0 confirms the solution because 0 < 8.
- Do not reverse an inequality when adding or subtracting.
- Do reverse it when multiplying or dividing by a negative.
- Always substitute an equation solution or test a value from an inequality region.
Worked examples
- 3(x + 2) = 21 → 3x + 6 = 21 → x = 5.
- 4x − 7 = 2x + 9 → 2x = 16 → x = 8.
- −3x ≥ 12 → x ≤ −4 (reverse the sign).
- 2(x − 1) + 5 = 11 → 2x + 3 = 11 → x = 4.
Independent practice
Try each question before revealing or checking the worked answer.
- Solve 5x − 4 = 21. Answer: x = 5.
- Solve 3(x + 2) = 24. Answer: x = 6.
- Solve 7x + 1 = 4x + 16. Answer: x = 5.
- Solve 2x − 5 < 9. Answer: x < 7.
- Solve −4x ≤ 20. Answer: x ≥ −5.
- Solve 6 − 2x > 12. Answer: x < −3.
- Solve 4(x − 1) = 2x + 10. Answer: x = 7.
- A number doubled and increased by 9 is 25. Answer: 2x + 9 = 25, x = 8.
Knowledge check
Choose one answer for each question, then check your answers. Explanations appear after checking.
Printable worksheet
Use the independent-practice questions above as your worksheet. Show each step, include units where relevant, and check answers only after attempting the questions.
Answer key
- Solve 5x − 4 = 21. Answer: x = 5.
- Solve 3(x + 2) = 24. Answer: x = 6.
- Solve 7x + 1 = 4x + 16. Answer: x = 5.
- Solve 2x − 5 < 9. Answer: x < 7.
- Solve −4x ≤ 20. Answer: x ≥ −5.
- Solve 6 − 2x > 12. Answer: x < −3.
- Solve 4(x − 1) = 2x + 10. Answer: x = 7.
- A number doubled and increased by 9 is 25. Answer: 2x + 9 = 25, x = 8.
Study tip: Write down the rule you used beside each answer. This makes errors easier to find and correct.