Q: What does IP Maths Notes (Lower Sec, Year 1-2): 02) Algebraic Expressions & Equations cover? A: Expand and factor expressions, solve linear equations and inequalities, and translate word problems into algebraic models.
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.
Group coefficients with identical literal parts (same variables and exponents). Ensure linear terms, constants, and powers stay distinct.
2. Factorisation toolkit
Common factor:6x2−9x=3x(2x−3).
Grouping:ax+ay+bx+by=(a+b)x+(a+b)y=(a+b)(x+y).
Difference of squares:a2−b2=(a+b)(a−b).
Perfect square trinomials:x2+6x+9=(x+3)2.
Worked example: Factorise 12y2−27.
12y2−27=3(4y2−9)=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⟹6x−5x=15+3⟹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⟹x<−3.
Represent solutions on a number line with open (strict) or closed (inclusive) dots.
3.3 Compound inequalities
For 2<3x+1≤11, isolate x:
2<3x+1≤11⟹1<3x≤10⟹31<x≤310.
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 unit of volume.
Tap rate: 301 per minute.
Drain rate: −451 per minute.
Net rate: 301−451=901 per minute.
Remaining volume: 0.6 unit.
Time required: t=9010.6=54 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.
Practice Quiz
Run through expansion, factorisation, balancing steps, and inequality checks with the interactive quiz before tackling the written set.
Try it yourself
Expand and simplify (x−4)(2x+5)−(x+3)2.
Factorise 15p2−21p.
Solve 35−2x=x−4.
Solve the inequality 2(3y−5)≤4−y and plot the solution on a number line.
Two numbers differ by 8 and their sum is 30. Form and solve simultaneous equations to find the numbers.
