Australian Curriculum v9 / ACiQ Year 8 Mathematics - Unit 2 - Linear equations and one-variable inequalities

Linear equations and one-variable inequalities

Solve linear equations and inequalities using inverse operations and represent inequality solutions correctly.

Updated 2026-07-26 - 12 min read

Linear equations and one-variable inequalities is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Solve multi-step linear equations

Maintain equality by applying the same inverse operation to both sides until the variable is isolated. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve multi-step linear equations and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The sign change is shorthand for applying an inverse operation to both sides; writing that operation preserves the logic.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Solve 3x + 7 = 25.

Step 1 - identify the governing idea: Maintain equality by applying the same inverse operation to both sides until the variable is isolated.

Step 2 - apply it to this evidence: Subtract 7 from both sides, then divide 18 by 3.

Result: 6

The relationship is visible in the working: Subtract 7 from both sides, then divide 18 by 3. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 8 — It does not agree with the required relationship: Subtract 7 from both sides, then divide 18 by 3.
  • 32/3 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Solve 5 − 2y = 17.

Step 1 - identify the governing idea: Maintain equality by applying the same inverse operation to both sides until the variable is isolated.

Step 2 - apply it to this evidence: Subtract 5 to get −2y = 12, then divide by −2.

Result: −6

The relationship is visible in the working: Subtract 5 to get −2y = 12, then divide by −2. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 6 — It does not agree with the required relationship: Subtract 5 to get −2y = 12, then divide by −2.
  • −11 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 11 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Solve 4(x − 3) = 2x + 10.

Step 1 - identify the governing idea: Maintain equality by applying the same inverse operation to both sides until the variable is isolated.

Step 2 - apply it to this evidence: Expanding gives 4x − 12 = 2x + 10; subtract 2x and add 12 to obtain 2x = 22.

Result: 11

The relationship is visible in the working: Expanding gives 4x − 12 = 2x + 10; subtract 2x and add 12 to obtain 2x = 22. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • −1 — It does not agree with the required relationship: Expanding gives 4x − 12 = 2x + 10; subtract 2x and add 12 to obtain 2x = 22.
  • 5.5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 17 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve one-variable inequalities

Use equation-like inverse operations, but reverse the inequality sign only when multiplying or dividing both sides by a negative number. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve one-variable inequalities and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Reversal is caused specifically by multiplying or dividing by a negative, because the order of values changes.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Solve 3x − 4 < 11.

Step 1 - identify the governing idea: Use equation-like inverse operations, but reverse the inequality sign only when multiplying or dividing both sides by a negative number.

Step 2 - apply it to this evidence: Add 4 and divide by positive 3, so the direction stays unchanged.

Result: x < 5

The relationship is visible in the working: Add 4 and divide by positive 3, so the direction stays unchanged. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x > 5 — It does not agree with the required relationship: Add 4 and divide by positive 3, so the direction stays unchanged.
  • x < 7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x = 5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Solve −2x ≤ 8.

Step 1 - identify the governing idea: Use equation-like inverse operations, but reverse the inequality sign only when multiplying or dividing both sides by a negative number.

Step 2 - apply it to this evidence: Dividing by −2 reverses ≤ to ≥.

Result: x ≥ −4

The relationship is visible in the working: Dividing by −2 reverses ≤ to ≥. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x ≤ −4 — It does not agree with the required relationship: Dividing by −2 reverses ≤ to ≥.
  • x ≥ 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x = −4 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

Solve 5 − x > 9.

Step 1 - identify the governing idea: Use equation-like inverse operations, but reverse the inequality sign only when multiplying or dividing both sides by a negative number.

Step 2 - apply it to this evidence: Subtract 5 to get −x > 4, then multiply by −1 and reverse the sign.

Result: x < −4

The relationship is visible in the working: Subtract 5 to get −x > 4, then multiply by −1 and reverse the sign. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • x > −4 — It does not agree with the required relationship: Subtract 5 to get −x > 4, then multiply by −1 and reverse the sign.
  • x < 4 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • x > 14 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Check and represent equation and inequality solutions

Substitute solutions into the original statement; graph inequalities with an open endpoint for < or > and a closed endpoint for ≤ or ≥. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by check and represent equation and inequality solutions and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: An inequality usually describes a set of values extending from a boundary, so it is represented by a ray or interval.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

Which value checks the equation 2x − 5 = 9?

Step 1 - identify the governing idea: Substitute solutions into the original statement; graph inequalities with an open endpoint for < or > and a closed endpoint for ≤ or ≥.

Step 2 - apply it to this evidence: Substitution gives 2(7) − 5 = 9.

Result: 7

The relationship is visible in the working: Substitution gives 2(7) − 5 = 9. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2 — It does not agree with the required relationship: Substitution gives 2(7) − 5 = 9.
  • 9 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 14 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

For x ≥ 3, what endpoint should a number-line graph use at 3?

Step 1 - identify the governing idea: Substitute solutions into the original statement; graph inequalities with an open endpoint for < or > and a closed endpoint for ≤ or ≥.

Step 2 - apply it to this evidence: The boundary value 3 is included because of the equals bar.

Result: A closed dot

The relationship is visible in the working: The boundary value 3 is included because of the equals bar. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • An open dot — It does not agree with the required relationship: The boundary value 3 is included because of the equals bar.
  • No endpoint — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • A cross — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

Which value satisfies −1 < x ≤ 4?

Step 1 - identify the governing idea: Substitute solutions into the original statement; graph inequalities with an open endpoint for < or > and a closed endpoint for ≤ or ≥.

Step 2 - apply it to this evidence: The upper boundary is included while the lower boundary −1 is excluded.

Result: 4

The relationship is visible in the working: The upper boundary is included while the lower boundary −1 is excluded. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • −1 — It does not agree with the required relationship: The upper boundary is included while the lower boundary −1 is excluded.
  • 5 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • −2 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Solve 5 − 2y = 17.
  2. Solve −2x ≤ 8.
  3. For x ≥ 3, what endpoint should a number-line graph use at 3?

Answers

  1. −6 — Subtract 5 to get −2y = 12, then divide by −2.
  2. x ≥ −4 — Dividing by −2 reverses ≤ to ≥.
  3. A closed dot — The boundary value 3 is included because of the equals bar.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources