🔢 Expressions, Brackets & Inequalities

Cambridge Lower Secondary Stage 7 · Unit 2

Writing Expressions
"5 more than x" → x + 5
"double y, minus 3" → 2y − 3
Expanding Brackets
3(2x + 4) = 6x + 12
Solving Inequalities
2x + 1 ≤ 9 → 2x ≤ 8 → x ≤ 4 ●————

What you'll learn:

  • Writing algebraic expressions from word descriptions
  • Expanding single brackets: a(b + c) = ab + ac
  • Expanding and simplifying two brackets
  • Solving inequalities (>, <, ≥, ≤)
  • Representing inequalities on a number line
  • Testing values to check inequalities
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📖 Learn: Expressions, Brackets & Inequalities

Part 1: Writing Algebraic Expressions

An algebraic expression uses letters (variables) to represent unknown numbers.

Words Expression
3 more than xx + 3
double y, minus 42y − 4
n divided by 5n ÷ 5
the square of mm²

Part 2: Expanding Single Brackets

The distributive law: multiply the term outside by each term inside.

a(b + c) = ab + ac

Example: 3(2x + 5) = 6x + 15

Algebra Tile Visualiser — click to animate!

Part 3: Expanding & Simplifying Two Brackets

Expand each bracket first, then collect like terms.

2(x + 3) + 3(x − 1)
= 2x + 6 + 3x − 3  (expand both brackets)
= 5x + 3  (collect like terms: 2x + 3x = 5x, 6 − 3 = 3)

Part 4: Inequalities

SymbolMeaningNumber Line
>greater thanopen circle, arrow right
<less thanopen circle, arrow left
≥greater than or equal toclosed circle, arrow right
≤less than or equal toclosed circle, arrow left

Solve an inequality the same way as an equation — but the inequality sign stays (flip it only if you multiply/divide by a negative, which we avoid at this level).

x + 3 > 7 → x > 4 (open circle at 4, arrow right)
2x ≤ 10 → x ≤ 5 (closed circle at 5, arrow left)

Part 5: Testing Values

To test whether a value satisfies an inequality, substitute and check.

Is x = 6 a solution to x + 3 > 7?   →   6 + 3 = 9 > 7 ✅ Yes!
Is x = 2 a solution to 2x ≤ 3?   →   2 × 2 = 4, is 4 ≤ 3? ❌ No!
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💡 Worked Examples

Example 1: Writing an Expression from Words

"A shop charges $p per book. Maya buys 4 books and gets a $3 discount. Write an expression for the total cost."

Step 1: Cost of 4 books = 4 × p = 4p
Step 2: Subtract the $3 discount.
Answer: 4p − 3

Example 2: Expanding a Single Bracket

Expand 5(3x − 2)

Step 1: Multiply 5 by 3x: 5 × 3x = 15x
Step 2: Multiply 5 by −2: 5 × (−2) = −10
Answer: 15x − 10
💡 Tip: Watch the sign! 5(3x − 2) ≠ 15x + 10

Example 3: Expanding and Simplifying Two Brackets

Expand and simplify 4(2x + 1) − 2(x − 3)

Step 1: Expand first bracket: 4 × 2x + 4 × 1 = 8x + 4
Step 2: Expand second bracket: −2 × x + (−2) × (−3) = −2x + 6
Step 3: Write together: 8x + 4 − 2x + 6
Step 4: Collect like terms: (8x − 2x) + (4 + 6) = 6x + 10

Example 4: Solving and Graphing an Inequality

Solve 3x − 1 > 8 and show it on a number line.

Step 1: Add 1 to both sides: 3x > 9
Step 2: Divide both sides by 3: x > 3
Step 3: Number line — open circle at 3, arrow pointing right.
0 1 2 3 4 5 6
Open circle (not equal to 3) · Arrow right (greater than 3)
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🔭 Bracket Expander Visualizer

Type a bracket expression like 3(2x + 5) or 4(x - 3), then click Expand!

Format: a(bx + c) or a(bx - c)  ·  e.g. 3(2x+5), 2(x-4), 5(3x+1)



Try these:

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✏️ Exercise 1: Expression Builder

Write an algebraic expression for each description. Use standard form (e.g. 2x+3, n-4).

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✏️ Exercise 2: Expanding Brackets

Expand each bracket. Fill in the blanks.

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✏️ Exercise 3: Expand and Simplify

Expand the brackets, then simplify by collecting like terms. Type the final simplified answer.

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✏️ Exercise 4: Solving Inequalities

Solve each inequality. Type your answer like x > 4 or x <= 3 (you can use <= for ≤ and >= for ≥).

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📏 Exercise 5: Number Line Plotter

For each inequality, click on the number line to place the circle, select open or closed, then click the arrow direction.

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✅ Exercise 6: True or False

Does the given value satisfy the inequality? Click TRUE or FALSE.

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📝 Practice Questions

  1. Write an expression for "7 more than x".
  2. Write an expression for "triple p, subtract 2".
  3. Write an expression for "half of n".
  4. Write an expression for "the sum of x and y, multiplied by 4".
  5. Expand: 2(x + 5)
  6. Expand: 3(2y − 4)
  7. Expand: 5(3x + 1)
  8. Expand: 4(a − 7)
  9. Expand: 6(2m + 3)
  10. Expand and simplify: 2(x + 3) + 4(x − 1)
  11. Expand and simplify: 3(2x + 5) + 2(x − 3)
  12. Expand and simplify: 5(x + 2) − 2(x − 1)
  13. Expand and simplify: 4(3x − 1) − 3(2x + 2)
  14. Solve: x + 4 > 9
  15. Solve: 2x < 14
  16. Solve: 3x − 1 ≥ 8
  17. Solve: 4x + 2 ≤ 18
  18. Solve: 5x − 3 > 12
  19. Is x = 5 a solution to x + 3 > 7? Show working.
  20. Is x = 2 a solution to 3x ≥ 8? Show working.
  1. x + 7
  2. 3p − 2
  3. n ÷ 2  (or n/2)
  4. 4(x + y)
  5. 2x + 10
  6. 6y − 12
  7. 15x + 5
  8. 4a − 28
  9. 12m + 18
  10. 2x + 6 + 4x − 4 = 6x + 2
  11. 6x + 15 + 2x − 6 = 8x + 9
  12. 5x + 10 − 2x + 2 = 3x + 12
  13. 12x − 4 − 6x − 6 = 6x − 10
  14. x > 5
  15. x < 7
  16. x ≥ 3
  17. x ≤ 4
  18. x > 3
  19. 5 + 3 = 8 > 7 ✅ Yes, x = 5 is a solution.
  20. 3 × 2 = 6, is 6 ≥ 8? ❌ No, x = 2 is not a solution.
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🏆 Challenge: Word Problems

  1. A cinema charges $d per ticket. Tom buys 3 tickets for himself and 2 friends and uses a $5 voucher. Write an expression for the total amount paid.
  2. A rectangular garden has length (2x + 3) m and width 4 m. Write an expression for the perimeter. Expand and simplify.
  3. A teacher has 5 boxes of pens. Each box contains (3p + 2) pens. Write and expand an expression for the total number of pens.
  4. Sam earns $(2x + 5) per hour and works 3 hours. He also earns a $10 bonus. Write and simplify an expression for his total earnings.
  5. Two classes each have (x + 8) students. Another class has 2(x − 3) students. Write and simplify an expression for the total number of students in all three classes.
  6. A bag of sweets costs $3. Emily has $m. She wants to buy more than 4 bags. Write an inequality and solve it to find the minimum value of m.
  7. A school bus can carry at most 48 passengers. Each row has 4 seats. There are r rows. Write and solve an inequality to find the maximum number of rows allowed.
  8. A phone plan costs $15 per month plus $0.10 per minute. Priya has a budget of at most $25 per month. Write and solve an inequality for the maximum number of minutes, m.
  9. A rectangle has width x cm and length (x + 5) cm. Its perimeter must be greater than 30 cm. Write and solve an inequality for x.
  10. Connor runs 2(x + 3) km on Monday and 3(x − 1) km on Wednesday. His weekly running goal is more than 25 km. Write and solve an inequality for x.
  1. Total = 3d − 5. (3 tickets at $d each, minus $5 voucher)
  2. Perimeter = 2 × (length + width) = 2(2x + 3 + 4) = 2(2x + 7) = 4x + 14 m.
  3. Total pens = 5(3p + 2) = 15p + 10.
  4. Earnings = 3(2x + 5) + 10 = 6x + 15 + 10 = 6x + 25.
  5. Total = (x + 8) + (x + 8) + 2(x − 3) = x + 8 + x + 8 + 2x − 6 = 4x + 10 students.
  6. 3 bags cost $3 × (number of bags). She buys more than 4 bags: 3n > 12. So n > 4, meaning m > 12, i.e. m > $12.
  7. 4r ≤ 48 → r ≤ 12. Maximum 12 rows.
  8. 15 + 0.10m ≤ 25 → 0.10m ≤ 10 → m ≤ 100. Maximum 100 minutes.
  9. Perimeter = 2x + 2(x + 5) = 2x + 2x + 10 = 4x + 10. Solve 4x + 10 > 30 → 4x > 20 → x > 5. So x > 5 cm.
  10. Total = 2(x + 3) + 3(x − 1) = 2x + 6 + 3x − 3 = 5x + 3. Solve 5x + 3 > 25 → 5x > 22 → x > 4.4. So x > 4.4 km.
Created by Mr Josh