IP Maths Notes (Lower Sec, Year 1-2): 02) Algebraic Expressions & Equations

Lower-sec algebra is a language: you model patterns, manipulate unknowns, and justify each line. This guide covers expansion, factorisation, equation solving, inequalities, and proportional reasoning.

Learning targets

• Expand binomials and collect like terms accurately.
• Factor expressions using common factors, grouping, and difference of squares.
• Solve single-variable linear equations and inequalities, showing balanced steps.
• Set up algebraic models from contextual problems (rates, mixtures, and ages).

1. Expansion and simplification

1.1 Distributive law refresher

For constants \( a, b, c \): \( a(b + c) = ab + ac \). Apply it to coefficents and like terms.

Tip: Always write intermediate steps when signs differ: \( -3(2x - 4) = -6x + 12 \).

1.2 Binomial products

Use FOIL (First, Outer, Inner, Last) or a tabular grid.

Example: Expand \( (2x - 3)(x + 5) \).

\[ (2x - 3)(x + 5) = 2x(x + 5) - 3(x + 5) = 2x^2 + 10x - 3x - 15 = 2x^2 + 7x - 15. \]

1.3 Collecting like terms

Group coefficients with identical literal parts (same variables and exponents). Ensure linear terms, constants, and powers stay distinct.

2. Factorisation toolkit

1. Common factor: \( 6x^2 - 9x = 3x(2x - 3) \).
2. Grouping: \( ax + ay + bx + by = (a + b)x + (a + b)y = (a + b)(x + y) \).
3. Difference of squares: \( a^2 - b^2 = (a + b)(a - b) \).
4. Perfect square trinomials: \( x^2 + 6x + 9 = (x + 3)^2 \).

Worked example: Factorise \( 12y^2 - 27 \).

\[ 12y^2 - 27 = 3(4y^2 - 9) = 3(2y + 3)(2y - 3). \]

3. Solving equations and inequalities

3.1 Linear equations

Maintain balance: whatever operation you apply to one side must be applied to the other.

Example: Solve \( 3(2x - 1) = 5(x + 3) \).

\[ 6x - 3 = 5x + 15 \implies 6x - 5x = 15 + 3 \implies x = 18. \]

Verify by substitution: LHS \( = 3(36 - 1) = 105 \), RHS \( = 5(21) = 105 \).

3.2 Inequalities

Flip the inequality sign only when multiplying or dividing by a negative number.

Example: Solve \( 4 - 3x > 13 \).

\[ -3x > 9 \implies x < -3. \]

Represent solutions on a number line with open (strict) or closed (inclusive) dots.

3.3 Compound inequalities

For \( 2 < 3x + 1 \leq 11 \), isolate \( x \):

\[ 2 < 3x + 1 \leq 11 \implies 1 < 3x \leq 10 \implies \frac{1}{3} < x \leq \frac{10}{3}. \]

4. Word problem modelling

Worked example — Rate mixture

A tap fills a tank in 30 minutes. A drain empties the same tank in 45 minutes. Both are opened together and the tank already contains 40% of its capacity. How long to fill the tank?

Let the tank capacity be \( 1 \text{ unit} \) of volume.

• Tap rate: \( \frac{1}{30} \) per minute.
• Drain rate: \( -\frac{1}{45} \) per minute.
• Net rate: \( \frac{1}{30} - \frac{1}{45} = \frac{1}{90} \) per minute.

Remaining volume: \( 0.6 \text{ unit} \).

Time required: \( t = \frac{0.6}{\frac{1}{90}} = 54 \text{ minutes} \).

Checklist for algebra word problems

• Define variables clearly (e.g., let \( x \) be the number of weeks, not the value of goods).
• Translate each sentence into an equation or inequality.
• Keep units consistent and annotate the meaning of intermediate expressions.

Try it yourself

1. Expand and simplify \( (x - 4)(2x + 5) - (x + 3)^2 \).
2. Factorise \( 15p^2 - 21p \).
3. Solve \( \frac{5 - 2x}{3} = x - 4 \).
4. Solve the inequality \( 2(3y - 5) \leq 4 - y \) and plot the solution on a number line.
5. Two numbers differ by 8 and their sum is 30. Form and solve simultaneous equations to find the numbers.

When ready, move on to https://eclatinstitute.sg/blog/ip-maths-lower-sec-notes/IP-Maths-Lower-Sec-03-Ratio-Rate-and-Percentage-Reasoning.