Practice Maths

Solving Linear Equations and Inequalities

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Key Terms

A linear equation has the form ax + b = c (no powers above 1). The solution is the value of x that makes it true.
The balance method: whatever you do to one side, you must do to the other — keeping the equation balanced.
Use inverse operations to isolate x: addition ↔ subtraction, multiplication ↔ division.
A linear inequality uses <, >, ≤, or ≥ instead of =. Solving steps are the same except: when multiplying or dividing by a negative number, the inequality sign flips.
On a number line: closed circle (•) means the endpoint IS included (≤, ≥); open circle (ˆ) means it is NOT included (<, >).
📚 Key Formulas and Rules
Solving equations: Isolate x using inverse operations. Always perform the same operation on both sides.
Equations with fractions: Multiply both sides by the LCM of all denominators to clear fractions.
Equations with brackets: Expand first, then collect like terms.
Inequality flip rule: If you multiply or divide both sides by a negative, flip the sign: < becomes >, ≤ becomes ≥, etc.
2x+3 11 Left side Right side 2x + 3 = 11 Both sides must stay balanced Number line examples: 0 1 2 3 4 x > 2 (open circle — 2 not included) 0 1 2 3 4 x ≤ 3 (closed circle — 3 IS included)
Hot Tip Remember the “flip the sign” rule for inequalities. Example: −2x < 6 → divide by −2 and flip: x > −3. This is the single most common error students make with inequalities.

Worked Example 1 — Solve an equation with brackets

Solve: 3(2x − 4) = 2x + 8

Step 1 — Expand: 6x − 12 = 2x + 8

Step 2 — Collect x terms: 6x − 2x = 8 + 12 → 4x = 20

Step 3 — Divide both sides by 4: x = 5

Check: LHS = 3(10−4) = 18; RHS = 10+8 = 18 ✓

Worked Example 2 — Solve an equation with fractions

Solve: x/3 + (x − 1)/2 = 4

Step 1 — Multiply through by LCM = 6: 6×(x/3) + 6×((x−1)/2) = 6×4

Step 2 — Simplify: 2x + 3(x−1) = 24 → 2x + 3x − 3 = 24

Step 3 — Collect: 5x = 27 → x = 27/5 = 5.4

Check: 5.4/3 + 4.4/2 = 1.8 + 2.2 = 4 ✓

Worked Example 3 — Solve and graph an inequality

Solve: 5 − 2x ≤ 11

Step 1 — Subtract 5 from both sides: −2x ≤ 6

Step 2 — Divide by −2 and FLIP the sign: x ≥ −3

Answer: x ≥ −3 (closed circle at −3, arrow to the right)

Introduction

Equations and inequalities appear constantly in real-world problems — from calculating costs and distances to engineering and finance. The ability to isolate an unknown variable is a fundamental mathematical skill. This lesson builds the systematic approach needed to solve any linear equation or inequality confidently.

The Balance Method

Think of an equation like a perfectly balanced set of scales. Whatever you do to one side, you must do exactly the same to the other side. The goal is to isolate the variable on one side by applying inverse operations in reverse order of BIDMAS.

Strategy: Solving step-by-step

Solve 4x − 7 = 13:

Step 1 — Deal with the constant first (add 7): 4x = 20

Step 2 — Deal with the coefficient (divide by 4): x = 5

Check: 4(5) − 7 = 20 − 7 = 13 ✓

Equations with Brackets

When an equation has brackets, expand first, then collect like terms, then isolate x.

Example: Brackets on both sides

Solve 2(3x + 1) = 4(x − 3) + 6:

Expand: 6x + 2 = 4x − 12 + 6 → 6x + 2 = 4x − 6

Collect x terms (subtract 4x): 2x + 2 = −6

Subtract 2: 2x = −8

Divide by 2: x = −4

Equations with Fractions

The cleanest approach is to multiply every term by the LCM of all denominators. This converts the equation into one with no fractions.

Example: Clearing fractions

Solve (2x)/3 − 1/4 = x/6:

LCM of 3, 4, 6 is 12. Multiply every term by 12:

12×(2x/3) − 12×(1/4) = 12×(x/6)

8x − 3 = 2x

Subtract 2x: 6x − 3 = 0 → 6x = 3 → x = 1/2

Linear Inequalities

Solving an inequality follows exactly the same steps as solving an equation, with one crucial exception: if you multiply or divide by a negative number, you must flip (reverse) the inequality sign.

Example: When to flip the sign

Solve −3x + 2 > 11:

Subtract 2: −3x > 9

Divide by −3 and FLIP (dividing by negative): x < −3

Number line: open circle at −3, shaded arrow pointing left

💡 Why does the sign flip? Consider 4 > 2. Multiply both sides by −1: −4 and −2. Is −4 > −2? No! −4 < −2. So the sign must flip to keep the statement true.

Word Problems — Setting Up Equations

To translate a word problem into an equation: identify the unknown, assign a variable, write an equation using the relationships described, then solve.

Example: Word problem

A plumber charges a $60 call-out fee plus $45 per hour. How many hours did they work if the total bill was $195?

Let h = hours worked.

Equation: 60 + 45h = 195

Subtract 60: 45h = 135

Divide by 45: h = 3 hours

Summary

Linear equations: use inverse operations to isolate x; expand brackets first; clear fractions using LCM; always check by substituting back. Linear inequalities: same process, but flip the sign when multiplying or dividing by a negative. On number lines: open circle for strict inequalities (<, >), closed circle for inclusive inequalities (≤, ≥).

Mastery Practice

See Answers ➔
  1. Fluency

    Solve each linear equation.

    1. (a) 3x + 7 = 22
    2. (b) 5x − 4 = 2x + 11
    3. (c) 8 − 3x = −1
    4. (d) −2x + 9 = 1
  2. Fluency

    Solve each equation involving brackets.

    1. (a) 2(x + 5) = 18
    2. (b) 3(2x − 1) = 5x + 9
    3. (c) 4(3 − x) = 2(x + 1)
    4. (d) −2(x − 4) = 3(2x + 1) − 7
  3. Fluency

    Solve each equation involving fractions.

    1. (a) x/4 + 3 = 7
    2. (b) x/2 − x/3 = 5
    3. (c) (x + 2)/3 = (2x − 1)/5
    4. (d) x/4 + (x − 3)/6 = 2
  4. Fluency

    Solve each inequality and represent the solution on a number line.

    1. (a) x + 4 > 9
    2. (b) 3x ≤ 12
    3. (c) 2x − 5 ≥ 7
    4. (d) −x + 3 < 8
  5. Understanding

    Solve each inequality (requiring sign flip) and describe the solution in words.

    1. (a) −4x < 20
    2. (b) 5 − 2x ≥ −3
    3. (c) −3(x + 2) ≤ 9
    4. (d) 2 − 5x > 17
  6. Understanding

    Solve each equation, expressing non-integer answers as fractions.

    1. (a) 3(x − 2) = 2(x + 1) + x − 4
    2. (b) (x + 1)/2 + (x − 3)/4 = 3
    3. (c) 2x/3 − (x + 1)/2 = 1/6
  7. Understanding

    For each situation, write an equation and solve it.

    1. (a) A cinema charges $12 per adult and $8 per child. A family pays $64 in total. If there are 2 adults, how many children attended?
    2. (b) The perimeter of a rectangle is 48 cm. The length is 3 cm more than twice the width. Find the dimensions.
  8. Understanding

    Identify and correct the error in each solution.

    1. (a) Solve −2x + 4 = 10. Student writes: −2x = 14, so x = −7. Is this correct? If not, what went wrong and what is the correct answer?
    2. (b) Solve −3x < 12. Student writes: x < −4. Is this correct? If not, what went wrong and what is the correct answer?
  9. Problem Solving

    Hiroshi has $240 to spend on a birthday party. The venue costs $80, and food costs $12.50 per person.

    1. (a) Write an inequality to represent the maximum number of people, n, Hiroshi can invite.
    2. (b) Solve the inequality to find the maximum number of guests.
    3. (c) If Hiroshi also wants to buy a birthday cake for $35, how does this change the maximum number of guests?
  10. Problem Solving

    Two friends, Priya and Sam, are each saving for a new phone.

    Priya has $150 saved and saves $25 per week. Sam has $80 saved and saves $40 per week.
    1. (a) Write an equation for each person’s total savings after w weeks.
    2. (b) After how many weeks will they both have the same amount saved? How much is that?
    3. (c) For what values of w does Sam have more than Priya? Write and solve an inequality.