Ch 4 · 4 Equations, Inequalities and Graphs · O Level Additional Mathematics 4037
Topic 4 of 6 in Algebra and Functions — O Level Additional Mathematics 4037 — 64 lessons.
Introduction
"The temperature must stay within 3 degrees of 20 °C" is the inequality |T - 20| ≤ 3. Modulus equations and inequalities like this, equations that become quadratics after a clever substitution, and the shapes of cubic graphs make up this topic. Graphs and algebra support each other: a sketch tells you how many solutions to expect. (Syllabus topic 4.)
Key ideas — Modulus equations
|A| = c (with c ≥ 0) means A = c or A = -c.
🧑🏫 Teacher's explanation
Think of the modulus symbol |...| as a 'distance machine'. It takes any number and tells you how far it is from zero on a number line, always returning a positive value. For example, |5| = 5 and |-5| = 5. Because distance cannot be negative, the equation |ax + b| = c only has solutions if c ≥ 0.
📖 Definitions
🔍 Worked example — Easy
Solve |2x - 5| = 7
🔍 Worked example — Standard
Solve |3x + 2| = x + 4
🔍 Worked example — Exam-style
Solve |x^2 - 5| = 4
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Modulus inequalities
Either method is accepted: algebraically, any valid method; graphically, the graph must be drawn accurately (on graph paper, to scale) so the solutions can be read from it.
🧑🏫 Teacher's explanation
The modulus of an expression, written as |A|, represents the distance of that expression from zero on a number line. Because distance cannot be negative, the modulus always outputs a value ≥ 0. Think of it like a 'distance sensor' on a car: if the sensor says the distance is 5 meters, the car could be 5 meters in front of you or 5 meters behind you. In algebra, this means |x| = 5 has two solutions: x = 5 and x = -5.
📖 Definitions
🔍 Worked example — Easy
Solve |2x - 5| < 3
🔍 Worked example — Exam-style
Solve |x^2 - 4| ≤ x + 2
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Substitution to make a quadratic
Spot a hidden quadratic and choose the substitution yourself:
🧑🏫 Teacher's explanation
Sometimes, an equation looks terrifying because it contains complicated expressions like √(x), ln(5x), or e^x instead of just a simple x. However, if you look closely, you might notice that the 'messy' part repeats itself. For example, in x - 5√(x) + 6 = 0, the term √(x) is the core component. If we pretend √(x) is just a single letter, say u, the equation transforms from a scary radical equation into a friendly quadratic equation: u^2 - 5u + 6 = 0.
📖 Definitions
🔍 Worked example — Standard
Solve 2(ln 5x)^2 + ln 5x - 6 = 0
🔍 Worked example — Exam-style
Solve 3e^x = 12 - 5e^-x
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Sketching cubics and their moduli
For y = (x - a)(x - b)(x - c) with a positive x^3 coefficient, the graph comes up from bottom left, crosses the x-axis at a, b and c, and goes off to top right (negative coefficient: the reverse). Label the x-intercepts and the y-intercept (put x = 0). A repeated factor (x - a)^2 means the curve touches the axis at a. For y = |f(x)|, reflect the parts below the axis.
🧑🏫 Teacher's explanation
Think of a cubic function like a roller coaster track. A standard cubic y = (x-a)(x-b)(x-c) doesn't just go in one direction; it snakes through the x-axis, crossing it at three distinct points. If the x^3 term is positive, you start deep underground (bottom left) and end high in the sky (top right). If there is a repeated factor, like (x-a)^2, the roller coaster doesn't cross the axis; it just 'kisses' it and bounces back in the same direction.
📖 Definitions
🔍 Worked example — Easy
Sketch the graph of y = |(x+2)(x-1)(x-3)|.
🔍 Worked example — Standard
Solve the equation |x^2 - 4| = 5.
🔍 Worked example — Exam-style
Solve the inequality |x-2| < x+4.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Cubic inequalities
To solve f(x) > d graphically: draw y = f(x) and the horizontal line y = d, find where they meet, and read off the x-values where the curve is above the line. For d = 0, simply use the roots and the sketch.
Worked examples — Example 1: modulus equation with a line
Solve |2x - 3| = x + 1.
🔍 Worked example — Easy
Solve |x - 5| = 3
🔍 Worked example — Standard
Solve |2x - 3| = x + 1
🔍 Worked example — Exam-style
Solve |3x + 2| = |x - 4|
⚠️ Common errors
Worked examples — Example 2: two moduli
Solve |x - 1| ≤ |2x + 3|. Square: x^2 - 2x + 1 ≤ 4x^2 + 12x + 9 ⇒ 3x^2 + 14x + 8 ≥ 0 ⇒ (3x + 2)(x + 4) ≥ 0, so x ≤ -4 or x ≥ -tfrac23.
🔍 Worked example — Standard
Solve |x - 1| ≤ |2x + 3|
🔍 Worked example — Exam-style
Solve |2x + 1| ≥ |x - 4|
⚠️ Common errors
Worked examples — Example 3: a quadratic in a modulus
Solve |x^2 - 4x| = 3. x^2 - 4x - 3 = 0 gives x = 2 ± sqrt7; x^2 - 4x + 3 = 0 gives x = 1, 3. Four solutions: 2 - sqrt7,\ 1,\ 3,\ 2 + sqrt7.
🔍 Worked example — Easy
Solve |x - 5| = 2
🔍 Worked example — Exam-style
Solve |x^2 - 5x| = 6
⚠️ Common errors
Worked examples — Example 4: substitution with logs
Solve 2(ln 5x)^2 + ln 5x - 6 = 0. With u = ln 5x: (2u - 3)(u + 2) = 0, so ln 5x = 1.5 or -2, giving x = tfrac15e^1.5 = 0.896 or x = tfrac15e^-2 = 0.0271.
🔍 Worked example — Easy
Solve x^2 - 5x + 6 = 0 using substitution.
🔍 Worked example — Exam-style
Solve 2(ln 5x)^2 + ln 5x - 6 = 0.
⚠️ Common errors
Worked examples — Example 5: a cubic and an inequality
f(x) = (x + 1)(x - 2)(x - 4) crosses the x-axis at -1, 2, 4 and the y-axis at 8. f(x) ≥ 0 for -1 ≤ x ≤ 2 or x ≥ 4.
🔍 Worked example — Easy
Given f(x) = (x + 1)(x - 2)(x - 4), find the y-intercept.
🔍 Worked example — Standard
Solve f(x) ≥ 0 for f(x) = (x + 1)(x - 2)(x - 4).
🔍 Worked example — Exam-style
Solve f(x) > 8 for f(x) = (x + 1)(x - 2)(x - 4).
⚠️ Common errors
Common mistakes
Practice
Answers: (1) x = 2 or x = -tfrac83. (2) |2x - 1| > 3: x > 2 or x < -1. (3) √(x) = 2 or 3: x = 4 or x = 9. (4) 3u^2 - 12u + 5 = 0, u = 2 ± √(21)/3: x = ln 3.528 = 1.26 or x = ln 0.472 = -0.750. (5) meets the x-axis at -2, 1, 3; y-intercept |(-1)(2)(-3)| = 6; parts below the axis (for x < -2 and 1 < x < 3) are reflected upwards.