Cambridge Lower Secondary CheckpointStage 9

Algebra (9As)

Mathematics Stage 9 Chapter Notes

What this chapter covers

Algebra - Sequences, functions and graphs
ShareWhatsAppPost
Algebra (9As) notes

Unable to load PDF

The notes viewer could not load. Please refresh the page.

Read online free. Download a watermarked copy with a free account.

Read the notes

The full Algebra (9As) notes as text: skim, search, and jump between subtopics.

~15 min read

In this chapter

  1. 1Simplifying Expressions & Index Laws
  2. 2Expanding Single and Double Brackets
  3. 3Factorising Algebraic Expressions
  4. 4Simplifying Algebraic Fractions
  5. 5Operations with Algebraic Fractions
  6. 6Solving Equations with Algebraic Fractions
  7. 7Solving Linear Inequalities

1. Simplifying Expressions & Index Laws

Simplifying an algebraic expression means making it shorter and easier to work with. The first step is to 'collect like terms'. Like terms are terms that have the exact same variables raised to the same powers. For example, in the expression `5x^2 + 2y - 3x^2`, the terms `5x^2` and `-3x^2` are like terms, and they simplify to `2x^2`. Index laws are rules that help us simplify expressions involving powers. When multiplying terms with the same base, you add the powers. When dividing, you subtract the powers. When raising a power to another power, you multiply them.

a^m × a^n = a^(m+n)

a^m ÷ a^n = a^(m-n)

(a^m)^n = a^(mn)

a^0 = 1 (for any non-zero a)

a^1 = a

Key term

Like Terms: Terms that have the exact same variables raised to the exact same powers.

Examiner insight

Examiners often test understanding of index laws with mixed variables, so ensure you only apply the laws to terms with the same base.

Common pitfall

A common error is to incorrectly add or subtract the bases, for example, writing `3x^2 + 2x^2 = 5x^4` instead of the correct `5x^2`.

Worked example 12 marks

Simplify the expression: `4x^2y + 3xy - 2x^2y + 5xy`

  1. 1

    Step 1: Identify the like terms. The `x^2y` terms are `4x^2y` and `-2x^2y`. The `xy` terms are `3xy` and `5xy`.

  2. 2

    Step 2: Group the like terms together: `(4x^2y - 2x^2y) + (3xy + 5xy)`.

  3. 3

    Step 3: Combine the coefficients of the like terms: `(4-2)x^2y + (3+5)xy`.

  4. 4

    Step 4: Write the final simplified expression: `2x^2y + 8xy`.

Worked example 22 marks

Simplify `(3p^4q^2) × (5p^2q^5)`

  1. 1

    Step 1: Multiply the numerical coefficients: `3 × 5 = 15`.

  2. 2

    Step 2: Apply the multiplication index law to the `p` terms: `p^4 × p^2 = p^(4+2) = p^6`.

  3. 3

    Step 3: Apply the multiplication index law to the `q` terms: `q^2 × q^5 = q^(2+5) = q^7`.

  4. 4

    Step 4: Combine all parts: `15p^6q^7`.

Worked example 32 marks

Simplify `(18a^9b^4) ÷ (6a^3b)`

  1. 1

    Step 1: Divide the numerical coefficients: `18 ÷ 6 = 3`.

  2. 2

    Step 2: Apply the division index law to the `a` terms: `a^9 ÷ a^3 = a^(9-3) = a^6`.

  3. 3

    Step 3: Apply the division index law to the `b` terms (remember `b` is `b^1`): `b^4 ÷ b^1 = b^(4-1) = b^3`.

  4. 4

    Step 4: Combine all parts: `3a^6b^3`.

Recap

  • To simplify, only combine terms with the exact same letters and powers.
  • When multiplying terms with the same base, add the indices.
  • When dividing terms with the same base, subtract the indices.
  • When raising a power to another power, multiply the indices.
  • Any base to the power of 0 is 1.

Quick check

  1. Simplify `2x^3 × 4x^5`.1 mark
  2. Simplify `10y^7 ÷ 2y^2`.1 mark

2. Expanding Single and Double Brackets

Expanding brackets means to multiply out the terms to remove the brackets. For a single bracket, you multiply the term outside by every term inside. For two brackets, like `(x+a)(x+b)`, you must multiply every term in the first bracket by every term in the second. A common method is FOIL: Firsts, Outers, Inners, Lasts. After expanding, always check if you can simplify by collecting like terms. Special cases include squaring a bracket, e.g. `(x+a)^2`, and the 'difference of two squares', `(x+a)(x-a)`.

a(b+c) = ab + ac

(a+b)(c+d) = ac + ad + bc + bd

(x+a)^2 = (x+a)(x+a) = x^2 + 2ax + a^2

(x-a)(x+a) = x^2 - a^2

Key term

Expand: To multiply out the terms in brackets to remove the brackets.

Examiner insight

Marks are often lost for sign errors, especially when expanding brackets involving negative terms. Double-check your multiplication, for example a negative times a negative is a positive.

Common pitfall

Forgetting the middle term when squaring a bracket, for example writing `(x+3)^2 = x^2 + 9` instead of `(x+3)^2 = x^2 + 6x + 9`.

Fun fact

The 'difference of two squares' pattern, `(x-a)(x+a) = x^2 - a^2`, is a powerful shortcut used in mental arithmetic, for example, `19 × 21 = (20-1)(20+1) = 400-1 = 399`.

Worked example 12 marks

Expand and simplify `(x + 6)(x - 4)`.

  1. 1

    Step 1 (Firsts): Multiply the first terms in each bracket: `x × x = x^2`.

  2. 2

    Step 2 (Outers): Multiply the outer terms: `x × -4 = -4x`.

  3. 3

    Step 3 (Inners): Multiply the inner terms: `6 × x = +6x`.

  4. 4

    Step 4 (Lasts): Multiply the last terms: `6 × -4 = -24`.

  5. 5

    Step 5: Write it all out and collect like terms: `x^2 - 4x + 6x - 24 = x^2 + 2x - 24`.

Worked example 22 marks

Expand and simplify `(2y - 5)^2`.

  1. 1

    Step 1: Write the expression as two brackets: `(2y - 5)(2y - 5)`.

  2. 2

    Step 2 (Firsts): `2y × 2y = 4y^2`.

  3. 3

    Step 3 (Outers): `2y × -5 = -10y`.

  4. 4

    Step 4 (Inners): `-5 × 2y = -10y`.

  5. 5

    Step 5 (Lasts): `-5 × -5 = +25`.

  6. 6

    Step 6: Combine and simplify: `4y^2 - 10y - 10y + 25 = 4y^2 - 20y + 25`.

Recap

  • To expand double brackets, multiply every term in the first bracket by every term in the second.
  • The FOIL method (First, Outer, Inner, Last) is a helpful reminder.
  • Always simplify by collecting like terms after expanding.
  • Remember that `(x+a)^2` expands to `x^2 + 2ax + a^2`, not just `x^2 + a^2`.
  • The expansion of `(x-a)(x+a)` is `x^2 - a^2` (difference of two squares).

Quick check

  1. Expand `3x(4-x)`.1 mark
  2. Expand `(x-5)(x+5)`.1 mark

3. Factorising Algebraic Expressions

Factorising is the reverse process of expanding brackets. It involves putting an expression back into brackets. The first type is finding the Highest Common Factor (HCF) of all terms and 'taking it outside' the bracket. The second main type is factorising a quadratic expression like `x^2 + bx + c` into two brackets, `(x+p)(x+q)`. To do this, you need to find two numbers, `p` and `q`, that multiply to give `c` and add to give `b`.

ax + ay = a(x+y)

x^2 + (p+q)x + pq = (x+p)(x+q)

Key term

Factorise: To express an algebraic expression as a product of its factors, usually involving brackets.

Examiner insight

When a question asks you to 'Factorise fully', it is a hint that there might be a common factor to take out first, or multiple steps involved.

Common pitfall

When factorising a quadratic like `x^2 - 5x + 6`, students often struggle with the signs. Remember that two negative numbers multiply to a positive and add to a negative, so the factors are `(x-2)(x-3)`.

Worked example 12 marks

Factorise fully `12x^2y - 18xy^3`.

  1. 1

    Step 1: Find the HCF of the numbers: HCF of 12 and 18 is 6.

  2. 2

    Step 2: Find the HCF of the `x` terms: HCF of `x^2` and `x` is `x`.

  3. 3

    Step 3: Find the HCF of the `y` terms: HCF of `y` and `y^3` is `y`.

  4. 4

    Step 4: The overall HCF is `6xy`. Place this outside the bracket: `6xy(...)`.

  5. 5

    Step 5: Divide the first term by the HCF: `12x^2y ÷ 6xy = 2x`.

  6. 6

    Step 6: Divide the second term by the HCF: `-18xy^3 ÷ 6xy = -3y^2`.

  7. 7

    Step 7: Write the final answer: `6xy(2x - 3y^2)`.

Worked example 22 marks

Factorise `x^2 + 8x + 15`.

  1. 1

    Step 1: We need two numbers that multiply to make +15 and add to make +8.

  2. 2

    Step 2: List pairs of factors of 15: (1, 15), (3, 5), (-1, -15), (-3, -5).

  3. 3

    Step 3: Check which pair adds to 8: `3 + 5 = 8`.

  4. 4

    Step 4: The numbers are +3 and +5. So the expression factorises to `(x+3)(x+5)`.

Worked example 32 marks

Factorise `y^2 - 3y - 10`.

  1. 1

    Step 1: We need two numbers that multiply to make -10 and add to make -3.

  2. 2

    Step 2: List pairs of factors of -10: (1, -10), (-1, 10), (2, -5), (-2, 5).

  3. 3

    Step 3: Check which pair adds to -3: `2 + (-5) = -3`.

  4. 4

    Step 4: The numbers are +2 and -5. So the expression factorises to `(y+2)(y-5)`.

Recap

  • Factorising is the opposite of expanding.
  • Always look for a Highest Common Factor first.
  • To factorise `x^2+bx+c`, find two numbers that multiply to `c` and add to `b`.
  • Pay close attention to positive and negative signs.
  • You can check your answer by expanding the brackets back out.

Quick check

  1. Factorise `15a^2 + 10a`.1 mark
  2. Factorise `x^2 - 9x + 20`.2 marks

4. Simplifying Algebraic Fractions

Simplifying algebraic fractions works just like simplifying numerical fractions: you divide the top (numerator) and bottom (denominator) by any common factors. The key step is often to factorise the numerator and denominator first. Once factorised, you can see the common factors (which are often brackets) and cancel them out. Remember, you can only cancel factors (things that are multiplied), not terms (things that are added or subtracted).

(ka) / (kb) = a / b

Key term

Cancel: In fractions, to divide both the numerator and the denominator by a common factor.

Examiner insight

Examiners look for the ability to factorise expressions correctly before simplifying the fraction. Show your factorisation step clearly to secure method marks.

Common pitfall

Incorrectly 'cancelling' terms that are added or subtracted, not multiplied. For example, you cannot cancel the `x` in `(x+2)/x` to get `(1+2)/1 = 3`. You can only cancel entire factors.

Worked example 12 marks

Simplify `(21a^5b^2) / (7a^2b^3)`

  1. 1

    Step 1: Simplify the numbers: `21 ÷ 7 = 3`.

  2. 2

    Step 2: Simplify the `a` terms using index laws: `a^5 ÷ a^2 = a^(5-2) = a^3`.

  3. 3

    Step 3: Simplify the `b` terms: `b^2 ÷ b^3 = b^(2-3) = b^-1`. This means `b` is left on the denominator.

  4. 4

    Step 4: Combine the results. The `3` and `a^3` are in the numerator, the `b` (`b^1`) is in the denominator. Answer: `(3a^3) / b`.

Worked example 23 marks

Simplify `(x^2 + 2x - 15) / (x^2 - 9)`

  1. 1

    Step 1: Factorise the numerator `x^2 + 2x - 15`. We need two numbers that multiply to -15 and add to +2. The numbers are +5 and -3. So, `x^2 + 2x - 15 = (x+5)(x-3)`.

  2. 2

    Step 2: Factorise the denominator `x^2 - 9`. This is a difference of two squares. So, `x^2 - 9 = (x+3)(x-3)`.

  3. 3

    Step 3: Rewrite the fraction with the factorised expressions: `((x+5)(x-3)) / ((x+3)(x-3))`.

  4. 4

    Step 4: Identify the common factor in the numerator and denominator, which is `(x-3)`. Cancel this factor.

  5. 5

    Step 5: Write the final simplified expression: `(x+5) / (x+3)`.

Recap

  • To simplify an algebraic fraction, first factorise the numerator.
  • Then, factorise the denominator.
  • Cancel any common factors that appear on both the top and bottom.
  • You can only cancel factors, not individual terms.

Quick check

  1. Can you simplify `(x+4)/(x+2)`? Explain why or why not.1 mark
  2. Simplify `(5(x+2)) / (10(x-1))`.1 mark

5. Operations with Algebraic Fractions

Performing operations (+, -, ×, ÷) on algebraic fractions follows the same rules as with numerical fractions. For multiplication, multiply the numerators together and the denominators together. For division, 'Keep, Flip, Change': keep the first fraction, flip the second upside down, and change the sign to multiply. For addition and subtraction, you must first find a common denominator. The easiest way is to multiply the two denominators together. Then, adjust the numerators accordingly before adding or subtracting them.

a/b × c/d = ac/bd

a/b ÷ c/d = a/b × d/c = ad/bc

a/b + c/d = (ad + bc) / bd

Key term

Common Denominator: A shared multiple of the denominators of two or more fractions, required for addition and subtraction.

Common pitfall

When subtracting fractions, forgetting to apply the negative sign to all terms in the second numerator. For example, in `(x+5)/2 - (x+1)/2`, the numerator becomes `x+5 - (x+1) = x+5 - x - 1 = 4`.

Worked example 13 marks

Simplify `(x+2)/5 × 10/(x^2-4)`

  1. 1

    Step 1: Factorise any parts of the fractions that can be factorised. `x^2 - 4` is a difference of two squares, `(x-2)(x+2)`.

  2. 2

    Step 2: Rewrite the expression: `(x+2)/5 × 10/((x-2)(x+2))`.

  3. 3

    Step 3: Multiply the numerators and denominators: `(10(x+2)) / (5(x-2)(x+2))`.

  4. 4

    Step 4: Cancel common factors. `(x+2)` is on the top and bottom. `10/5` simplifies to `2`.

  5. 5

    Step 5: The final answer is `2 / (x-2)`.

Worked example 23 marks

Write as a single fraction: `3/(x-1) + 4/(x+2)`

  1. 1

    Step 1: Find a common denominator by multiplying the two denominators: `(x-1)(x+2)`.

  2. 2

    Step 2: Rewrite the first fraction with the new denominator. We multiplied the bottom by `(x+2)`, so we must multiply the top by `(x+2)`: `3(x+2) / ((x-1)(x+2))`.

  3. 3

    Step 3: Rewrite the second fraction. We multiplied the bottom by `(x-1)`, so we must multiply the top by `(x-1)`: `4(x-1) / ((x-1)(x+2))`.

  4. 4

    Step 4: Add the numerators over the common denominator: `(3(x+2) + 4(x-1)) / ((x-1)(x+2))`.

  5. 5

    Step 5: Expand and simplify the numerator: `(3x + 6 + 4x - 4) / ((x-1)(x+2)) = (7x + 2) / ((x-1)(x+2))`.

Recap

  • To multiply fractions, multiply the tops and multiply the bottoms. Simplify first if possible.
  • To divide by a fraction, flip it and multiply.
  • To add or subtract fractions, you must have a common denominator.
  • Always simplify your final answer by cancelling common factors if possible.

Quick check

  1. Calculate `(x/3) × (6/y)`.1 mark
  2. What is the common denominator for `2/x` and `5/(x+1)`?1 mark

6. Solving Equations with Algebraic Fractions

When an equation contains fractions with the unknown variable in the denominator, the goal is to eliminate the fractions. This is done by multiplying every single term in the equation by the Lowest Common Denominator (LCD). This process is often called 'clearing the denominators'. Once the fractions are gone, you will be left with a standard linear or quadratic equation that you already know how to solve.

If a/x = b, then a = bx.

If a/x + b/y = c, multiply all terms by xy: ay + bx = cxy.

Key term

Clear the Denominator: A method for solving fractional equations by multiplying every term by the lowest common multiple of the denominators.

Examiner insight

Show the step where you multiply all terms by the common denominator. This demonstrates your method clearly and can earn marks even if you make a subsequent calculation error.

Common pitfall

Forgetting to multiply terms that are not fractions by the common denominator. Every single term on both sides of the equals sign must be multiplied.

Worked example 13 marks

Solve the equation `6/(x+2) = 2`.

  1. 1

    Step 1: The only denominator is `(x+2)`. Multiply both sides of the equation by `(x+2)`.

  2. 2

    Step 2: `6/(x+2) × (x+2) = 2 × (x+2)`.

  3. 3

    Step 3: This simplifies to `6 = 2(x+2)`.

  4. 4

    Step 4: Expand the bracket: `6 = 2x + 4`.

  5. 5

    Step 5: Solve the linear equation. Subtract 4 from both sides: `2 = 2x`.

  6. 6

    Step 6: Divide by 2: `x = 1`.

Worked example 23 marks

Solve the equation `5/x + 3/2 = 4`.

  1. 1

    Step 1: Identify the denominators: `x` and `2`. The Lowest Common Denominator (LCD) is `2x`.

  2. 2

    Step 2: Multiply every term in the equation by `2x`.

  3. 3

    Step 3: `(5/x)×2x + (3/2)×2x = 4×2x`.

  4. 4

    Step 4: Simplify each term. In the first term, the `x`'s cancel, leaving `5×2=10`. In the second term, the `2`'s cancel, leaving `3×x=3x`. The right side becomes `8x`.

  5. 5

    Step 5: The equation is now `10 + 3x = 8x`.

  6. 6

    Step 6: Solve the equation. Subtract `3x` from both sides: `10 = 5x`.

  7. 7

    Step 7: Divide by 5: `x = 2`.

Recap

  • To solve an equation with fractions, find the Lowest Common Denominator (LCD).
  • Multiply every term on both sides of the equation by the LCD.
  • This 'clears' the denominators, leaving a simpler equation.
  • Solve the resulting linear or quadratic equation.
  • Always check your answer by substituting it back into the original equation.

Quick check

  1. What would you multiply by to solve `4/(x-1) + 3/x = 2`?1 mark

7. Solving Linear Inequalities

Solving a linear inequality is very similar to solving a linear equation. You can add, subtract, multiply, or divide both sides by a value to isolate the variable. There is one crucial rule to remember: if you multiply or divide both sides of an inequality by a negative number, you must flip the direction of the inequality sign. For example, `<` becomes `>` and `≥` becomes `≤`. Solutions can be represented on a number line, using an open circle for `<` and `>` (the value is not included) and a closed circle for `≤` and `≥` (the value is included).

If a < b and c > 0, then ac < bc and a/c < b/c.

If a < b and c < 0, then ac > bc and a/c > b/c. (Note the flipped sign)

Key term

Inequality: A mathematical statement that compares two values that are not equal, using symbols like <, >, ≤, or ≥.

Examiner insight

Examiners often test the 'flip the sign' rule. Be alert for questions that require you to divide by a negative coefficient, such as solving `5 - 2x > 1`.

Common pitfall

The most common mistake is forgetting to reverse the inequality symbol when multiplying or dividing by a negative number.

Worked example 12 marks

Solve the inequality `4x - 5 > 11` and show the solution on a number line.

  1. 1

    Step 1: Treat it like an equation. Add 5 to both sides: `4x > 11 + 5`.

  2. 2

    Step 2: Simplify: `4x > 16`.

  3. 3

    Step 3: Divide both sides by 4. Since 4 is positive, the sign does not flip: `x > 4`.

  4. 4

    Step 4: To show on a number line, draw a line with the number 4. Place an open circle at 4 (because it's `>` not `≥`) and draw an arrow pointing to the right (for all numbers greater than 4).

Worked example 22 marks

Solve the inequality `8 - 3x ≥ 17`.

  1. 1

    Step 1: Subtract 8 from both sides: `-3x ≥ 17 - 8`.

  2. 2

    Step 2: Simplify: `-3x ≥ 9`.

  3. 3

    Step 3: Divide both sides by -3. Because we are dividing by a negative number, we must flip the inequality sign.

  4. 4

    Step 4: `x ≤ 9 / -3` which simplifies to `x ≤ -3`.

Worked example 33 marks

Find the largest integer that satisfies the inequality `5(x-2) < 18`.

  1. 1

    Step 1: Expand the bracket: `5x - 10 < 18`.

  2. 2

    Step 2: Add 10 to both sides: `5x < 28`.

  3. 3

    Step 3: Divide by 5: `x < 5.6`.

  4. 4

    Step 4: The question asks for the largest integer value of x. The integers less than 5.6 are 5, 4, 3, ...

  5. 5

    Step 5: The largest of these integers is 5.

Recap

  • Solve inequalities like you solve equations: do the same thing to both sides.
  • If you multiply or divide both sides by a negative number, you MUST flip the inequality sign.
  • Use an open circle on a number line for < and >.
  • Use a closed circle on a number line for ≤ and ≥.
  • If asked for an integer solution, find the specific whole number that satisfies the condition.

Quick check

  1. Solve `-2x > 10`.1 mark
  2. What is the smallest integer that satisfies `x ≥ 3.5`?1 mark

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Simplify fully: `(2x^4y)^3`2 marks
  2. Expand and simplify: `(3x - 2)(x + 5)`2 marks
  3. Factorise completely: `2x^2 - 18`3 marks
  4. Solve the inequality `15 - 4x > 7`2 marks
  5. Simplify fully: `(x^2 - 25) / (x^2 + 7x + 10)`3 marks
  6. Write as a single fraction in its simplest form: `4/(x-3) - 5/(x+1)`3 marks
  7. Solve the equation: `7/(x+2) + 1 = 10/x`4 marks
  8. The area of a rectangle is `x^2 + 5x - 6`. The width of the rectangle is `x-1`. Find an expression for the length of the rectangle.3 marks
  9. Find the set of integer values for which `21 < 6x - 3 ≤ 39`.3 marks
  10. A triangle has sides of length `x`, `x+3` and `x+5`. The perimeter of the triangle is greater than 20cm. Find the smallest possible integer value for x.3 marks

Go deeper

Practise and revise with member-only material for this chapter.

Workbook

Exam-style practice questions with full worked solutions.

Unlock with a plan

Chapter at a Glance

A single-page summary of every key idea for fast revision.

Unlock with a plan

Free notes are just the start.

Unlock every Workbook and Chapter at a Glance, and generate your own worksheets and predicted papers.

Explore plans

Related chapters

Number (9Ni)Number (9Np)Number (9Nf)Algebra (9Ae)Geometry and Measure (9Gg)Geometry and Measure (9Gp)