Algebra, Equations & Inequalities
Table of Contents
Algebra, Equations & Inequalities (~25 marks, Paper 1)#
This section is unusual — most of its marks come from Grade 10 and 11 content, but it’s tested in your matric exam. Question 1 of Paper 1 is almost always a set of “algebra warm-up” questions that draw on everything you’ve learned since Grade 10, sometimes mixed with Grade 12 concepts like the Factor Theorem or logarithms.
The good news: If you’ve been doing maths for 3 years, this should be your easiest 25 marks. The bad news: Students who haven’t revised their basics lose marks here that they can never recover.
What Typically Appears in Question 1#
Based on past papers, here’s what you can expect:
| Question Type | Grade Level | Typical Form |
|---|---|---|
| Linear equation | Gr 10 | Solve for $x$: $3(x-2) = 5x + 4$ |
| Quadratic equation (factorise) | Gr 11 | Solve for $x$: $x^2 - 5x + 6 = 0$ |
| Quadratic equation (formula) | Gr 11 | Solve for $x$: $2x^2 + 3x - 7 = 0$ (correct to 2 decimal places) |
| Simultaneous equations | Gr 10–11 | Solve for $x$ and $y$: $y = 2x - 1$ and $x^2 + y^2 = 5$ |
| Quadratic inequality | Gr 11 | Solve for $x$: $x^2 - 4x - 5 \leq 0$ |
| Nature of roots (discriminant) | Gr 11 | For which values of $k$ will $x^2 + kx + 9 = 0$ have real roots? |
| Surd equation | Gr 11 | Solve for $x$: $\sqrt{x+3} = x - 3$ |
| Exponential equation | Gr 11 | Solve for $x$: $3^{x+1} = 27$ |
| Logarithmic equation | Gr 12 | Solve for $x$: $\log_2(x-1) + \log_2(x+1) = 3$ |
| Factor Theorem application | Gr 12 | Solve for $x$: $x^3 - 7x - 6 = 0$ |
Quick Revision: The Key Skills#
1. Linear Equations#
Undo operations in reverse order. Always check your answer by substituting back.
$3(x - 2) = 5x + 4$
$3x - 6 = 5x + 4$
$-10 = 2x$
$x = -5$
2. Quadratic Equations#
Method 1: Factorise (when possible)
$x^2 - 5x + 6 = 0$
$(x-2)(x-3) = 0$
$x = 2$ or $x = 3$
Method 2: Quadratic Formula (always works)
$$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$For $2x^2 + 3x - 7 = 0$: $a = 2$, $b = 3$, $c = -7$
$x = \frac{-3 \pm \sqrt{9 + 56}}{4} = \frac{-3 \pm \sqrt{65}}{4}$
$x = 1.27$ or $x = -2.77$ (to 2 decimal places)
3. Simultaneous Equations#
Strategy: Substitute the linear equation into the quadratic one.
$y = 2x - 1$ … (1)
$x^2 + y^2 = 5$ … (2)
Substitute (1) into (2):
$x^2 + (2x-1)^2 = 5$
$x^2 + 4x^2 - 4x + 1 = 5$
$5x^2 - 4x - 4 = 0$
$(5x + 4)(x - 1) = 0 \quad \Rightarrow \quad x = -\frac{4}{5}$ or $x = 1$
Then find $y$ from equation (1) for each $x$.
4. The Discriminant ($\Delta$)#
For $ax^2 + bx + c = 0$:
$$ \Delta = b^2 - 4ac $$| Value of $\Delta$ | Nature of roots |
|---|---|
| $\Delta > 0$ and a perfect square | Two real, rational, unequal roots |
| $\Delta > 0$ and NOT a perfect square | Two real, irrational, unequal roots |
| $\Delta = 0$ | Two equal (repeated) real roots |
| $\Delta < 0$ | No real roots (non-real/imaginary) |
Example: Finding $k$#
For which values of $k$ will $x^2 + kx + 9 = 0$ have equal roots?
$\Delta = 0$: $k^2 - 4(1)(9) = 0 \Rightarrow k^2 = 36 \Rightarrow k = \pm 6$
For real roots: $\Delta \geq 0$: $k^2 - 36 \geq 0 \Rightarrow k \leq -6$ or $k \geq 6$
5. Quadratic Inequalities#
$x^2 - 4x - 5 \leq 0$
Step 1: Factorise: $(x-5)(x+1) \leq 0$
Step 2: Critical values: $x = 5$ and $x = -1$
Step 3: Sign diagram or sketch a mini-parabola:
The parabola opens upward (positive $x^2$), so it’s negative BETWEEN the roots:
$-1 \leq x \leq 5$
6. Surd Equations#
$\sqrt{x + 3} = x - 3$
Step 1: Square both sides: $x + 3 = (x-3)^2 = x^2 - 6x + 9$
Step 2: Rearrange: $x^2 - 7x + 6 = 0 \Rightarrow (x-1)(x-6) = 0$
Step 3: $x = 1$ or $x = 6$
Step 4: CHECK both in the original! (Squaring can introduce false solutions.)
$x = 1$: $\sqrt{4} = -2$? NO. Rejected.
$x = 6$: $\sqrt{9} = 3$? YES. ✓
Grade 12 Additions to This Section#
While most questions are revision, some questions bring in Grade 12 concepts:
- Solving cubic equations using the Factor Theorem → See Polynomials: Solving Cubics
- Logarithmic equations → See Functions: Logarithmic Function
- Nature of roots with parameters — these get harder in Grade 12 but use the same discriminant logic
Build on Your Lower-Grade Foundations#
All the skills in this section are taught in detail at the grade level where they first appear. If any feel shaky, go back to the source:
- Grade 10: Algebraic Expressions — Expanding brackets, factoring, DOTS, trinomials
- Grade 10: Equations & Inequalities — Linear equations, literal equations, simultaneous equations
- Grade 10: Exponents — Laws of exponents, simplifying
- Grade 11: Equations & Inequalities — Quadratic formula, discriminant, quadratic inequalities, simultaneous with quadratics
- Grade 11: Exponents & Surds — Surd equations, rationalising
🚨 Common Mistakes#
- Not checking surd equation answers: Squaring both sides can create false solutions. ALWAYS substitute back into the original.
- Discriminant sign errors: $\Delta = b^2 - 4ac$. Make sure you use the correct sign for $c$. If $c$ is negative, then $-4ac$ becomes positive.
- Simultaneous equations — expanding errors: When substituting $y = 2x - 1$ into $y^2$, you must expand $(2x-1)^2$ correctly: $4x^2 - 4x + 1$, NOT $4x^2 - 1$.
- Inequality sign flip: If you multiply or divide by a negative, the inequality sign REVERSES.
- Leaving answers as decimals when they should be exact: If the question says “solve”, give exact answers (fractions, surds). Only round if asked for “correct to 2 decimal places”.
💡 Pro Tip: Question 1 = Free Marks#
This is the most predictable section of the exam. The question types repeat almost identically year after year. Do every Question 1 from the past 5 years and you’ll walk into the exam knowing exactly what to expect.
⏮️ Fundamentals | 🏠 Back to Grade 12 | ⏭️ Sequences & Series
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