VCE Vault

Year 9 Mathematics · Number and Algebra · 18m read

Linear Equations and Inequalities

Ever wondered why some problems have one exact answer while others give you a whole range of possibilities? That's the difference between equations (finding that one magic number) and inequalities (discovering all the numbers that work). You'll learn to solve both and translate real-world situations—like budgeting or speed limits—into mathematical language. These skills are essential for VCE maths and deciding everything from "Can I afford this?" to "What's the safe range for this measurement?"

Learning objective: Solve linear equations, inequalities, and word problems.

By the end of this lesson you can:

  • I solve multi-step linear equations.
  • I solve and graph inequalities.
  • I translate word problems into equations.

Equations balance; inequalities tip. An equation is a level seesaw — keep it level by doing the same to both sides. An inequality is a tipped seesaw, and multiplying by a negative flips which side is heavier.

Balance and inequality

LINEAR EQUATIONS — the balance method.

The equals sign means BALANCE. Whatever you do to one side, do the same to the other.

### Standard Procedure
1. Expand brackets.
2. Collect like terms.
3. Get variable terms on ONE side.
4. Get constant terms on the OTHER side.
5. Divide by the coefficient.

Example (multi-step):
$3(x - 2) = 15$
$3x - 6 = 15$ (expand)
$3x = 21$ (add 6 to both sides)
$x = 7$ (divide by 3)
Check: $3(7 - 2) = 3 \times 5 = 15$ ✓

### Equations with Variables on Both Sides
$5x + 3 = 2x + 12$
$3x = 9$ (subtract $2x$ and $3$)
$x = 3$

### Equations with Fractions
$\frac{x}{3} + 2 = 5$
Multiply both sides by 3: $x + 6 = 15$
$x = 9$

For $\frac{2x+1}{4} = 3$: multiply both sides by 4: $2x + 1 = 12 \Rightarrow x = 5.5$

### INEQUALITIES
Same rules as equations, EXCEPT:
Multiply or divide by a NEGATIVE → FLIP the inequality sign.

Symbols and their meaning:
• $x > 3$ means $x$ is strictly greater than 3.
• $x \geq 3$ means $x$ is 3 or greater.
• $x < -1$ means $x$ is strictly less than $-1$.

Example: $-2x > 6$
Divide both sides by $-2$ (FLIP): $x < -3$

### Graphing on Number Lines
• Open circle $\circ$ for strict inequalities ($<$ or $>$).
• Filled circle $\bullet$ for $\leq$ or $\geq$.
• Arrow pointing right for $x > a$, left for $x < a$.

### Compound Inequalities
$-3 \leq x < 5$: values between $-3$ (included) and $5$ (excluded).

### Word Problems — 5-step method
1. IDENTIFY what you are finding (let $x = ...$).
2. TRANSLATE words into algebra.
3. FORM the equation/inequality.
4. SOLVE.
5. INTERPRET: write the answer in context with units and check it makes sense.

### Australian context
A plumber charges $80 plus $60 per hour. To keep the bill under $380 (inequality): $80 + 60h < 380 \Rightarrow h < 5$.

Key Relationships

  • General linear equation: ax + b = c \Rightarrow x = \dfrac{c-b}{a}
  • Flip rule: \text{If } -ax > b \text{ then } x < -\dfrac{b}{a} (Divide by negative → flip sign)

Equation watchlist

  • Flip on negative: ONLY for inequalities when dividing/multiplying by a negative number.
  • Sub back in: Always verify your answer by substitution.
  • Units: Include units in the final answer for word problems.

Quick reference

| Example | Solution |
|---|---|
| 5x − 7 = 18 | x = 5 |
| 2(x + 3) = 4x − 2 | x = 4 |
| (x + 1)/3 = 4 | x = 11 |
| 3 − 2x ≥ 9 | x ≤ −3 (flip) |
| Number line | Open circle < >, closed ≤ ≥ |

Worked example (Medium)

Twice a number increased by 7 is 23. Find the number.

  1. Let. Let $x$ = the number.
  2. Equation. $2x + 7 = 23$
  3. Solve. $2x = 16$ (subtract 7); $x = 8$ (divide by 2).
  4. Check. $2(8) + 7 = 16 + 7 = 23$ ✓

Answer: The number is 8.

Worked example (Hard)

Solve $\dfrac{3(x-2)}{5} - \dfrac{x+1}{2} = 1$.

  1. LCM. Multiply every term by 10 (LCM of 5 and 2): $2 \cdot 3(x-2) - 5(x+1) = 10$
  2. Expand. $6x - 12 - 5x - 5 = 10$
  3. Simplify. $x - 17 = 10$
  4. Solve. $x = 27$
  5. Check. $\dfrac{3(25)}{5} - \dfrac{28}{2} = 15 - 14 = 1$ ✓

Answer: $x = 27$

Common mistakes

  • Solving $-3x < 12$ as $x < -4$ (not flipping the sign) When dividing both sides by $-3$ (a negative), the inequality MUST flip direction. Fix: $-3x < 12 \Rightarrow x > -4$ (flip from $<$ to $>$). Check: $x = 0$: $-3(0) = 0 < 12$ ✓, and $0 > -4$ ✓.
  • Writing $x = 5$ when the answer to a word problem should be in context Equations are tools; the answer must be interpreted in the real situation. Fix: Write: "The cost is $x = 5$ items" or "The time is $t = 3$ hours". Include the unit and noun.
  • Adding $x$ to only one side when $x$ appears on both sides The balance principle requires every operation to be applied to BOTH sides. Fix: Write each step fully: from $5x = 2x + 9$, subtract $2x$ from both sides: $3x = 9$.

Practice questions

  1. Solve: $2x - 7 = 13$

    1. $x = 3$
    2. $x = 5$
    3. $x = 10$
    4. $x = 7$

    (1 marks)

  2. Solve $-4x > 12$.

    (2 marks)

  3. Solve $4(2x - 1) = 3(x + 5)$.

    (2 marks)

  4. A gym membership costs $\$50$ joining fee plus $\$30$ per month. Another gym charges $\$20$ per month but a $\$110$ fee. For how many months is the first gym cheaper?

    (3 marks)

  5. The perimeter of a rectangle is 46 cm. The length is 5 cm more than the width. Find the dimensions.

    (3 marks)

Going Deeper — Equations from Geometry

Many geometry problems reduce to linear equations. Here are three classic types:

TYPE 1 — Angles in a triangle:
Three angles sum to $180°$. If angles are $x$, $2x$, and $x + 30$, then:
$x + 2x + x + 30 = 180 \Rightarrow 4x = 150 \Rightarrow x = 37.5°$

TYPE 2 — Parallel lines with transversal:
Co-interior angles sum to $180°$. If angles are $3x + 10$ and $2x + 20$:
$3x + 10 + 2x + 20 = 180 \Rightarrow 5x = 150 \Rightarrow x = 30$

TYPE 3 — Equal segments:
A line $AB$ is 30 cm. $C$ lies on $AB$ with $AC = 2x + 1$ and $CB = x + 5$:
$2x + 1 + x + 5 = 30 \Rightarrow 3x = 24 \Rightarrow x = 8$

These appear frequently in NAPLAN and state exam papers — recognise the "sum = total" structure.

Summary

  • Balance method: apply the same operation to BOTH sides.
  • Standard order: expand → collect → variables one side → constants other → divide.
  • Inequalities: identical rules EXCEPT dividing/multiplying by a negative FLIPS the sign.
  • Number line: open circle for strict ($< >$), filled for $\leq \geq$.
  • Word problems: define variable → form equation → solve → interpret with units → check.

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