Ch 4 · 2.6 Inequalities · O Level Mathematics 4024
Topic 4 of 8 in Algebra and Graphs — O Level Mathematics 4024 — 48 lessons.
Introduction
"You must be at least 120 cm tall to ride", "the bag must weigh less than 23 kg": real limits are inequalities. This lesson shows them on number lines, solves them like equations (with one extra rule), and draws regions on a graph that satisfy several inequalities at once. (Syllabus section 2.6. Linear programming is not included.)
Key ideas — Symbols and number lines
x < 3 (less than), x ≤ 3 (less than or equal to), x > -1, x ≥ -1. On a number line:
🧑🏫 Teacher's explanation
Think of an inequality as a 'boundary' or a 'limit' rather than a single fixed point. If I say, "You must be at least 12 years old to ride this rollercoaster," I am setting a limit. Anyone exactly 12 is allowed (inclusive), but anyone 11 is not. In maths, we use symbols to describe these limits. Instead of saying x = 3, which is just one specific point, we say x > 3, which describes an infinite range of numbers.
📖 Definitions
🔍 Worked example — Easy
Represent the inequality -2 < x ≤ 3 on a number line.
🔍 Worked example — Standard
Solve the inequality 3x - 5 < x + 7 and represent the solution on a number line.
🔍 Worked example — Exam-style
Solve the compound inequality -3 ≤ 3x - 2 < 7.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving linear inequalities
Solve like an equation, with one extra rule: multiplying or dividing by a negative number reverses the inequality.
🧑🏫 Teacher's explanation
Solving an inequality is almost exactly like solving a normal equation (with an '=' sign), but instead of finding one single magic number, you are finding a whole range of numbers that make the statement true. Think of an equation like a single point on a map, whereas an inequality is like a territory or a zone. If I say 'you must be older than 18', I'm not looking for one specific age; I'm looking for a whole group of people.
📖 Definitions
🔍 Worked example — Easy
Solve the inequality 3x < 2x + 4 and represent the solution on a number line.
🔍 Worked example — Standard
Solve the compound inequality -3 ≤ 3x - 2 < 7.
🔍 Worked example — Exam-style
Solve 5 - 2x > 11 and state the direction of the inequality sign.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Regions on a graph
For an inequality in x and y, draw the boundary line first:
🧑🏫 Teacher's explanation
Think of an inequality like a boundary fence. A standard equation like y = 2x is a thin, precise line. An inequality like y > 2x is like saying, "You can stand anywhere on one side of this fence."
📖 Definitions
🔍 Worked example — Easy
Represent x > 3 on a number line.
🔍 Worked example — Standard
Represent y ≤ 2x + 1 on a graph.
🔍 Worked example — Exam-style
Shade the region satisfying x + y < 4 and y > 1.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Finding the inequalities for a region
Find the equation of each boundary line, then decide the direction by testing a point inside the region, and whether the line is solid or broken.
🧑🏫 Teacher's explanation
Think of an inequality as a boundary fence in a field. A single equation like y = 2x + 1 is like a thin wire fence; it tells you exactly where the edge is. An inequality like y > 2x + 1 tells you that you can stand anywhere on one side of that fence, but not on the fence itself.
📖 Definitions
🔍 Worked example — Easy
Find the inequality represented by a solid line passing through (0, 2) and (2, 6), where the region is above the line.
🔍 Worked example — Standard
A region is bounded by the line x + y = 5. The region includes the line and is below it. Write the inequality.
🔍 Worked example — Exam-style
A region is defined by the intersection of y > 2 and x < 3. Describe the boundary lines and the type of lines used.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Sketch the region on a coordinate plane, using appropriate line types. (b) List the inequalities that define this region. [4 marks]
Worked examples
Find the integers n with -4 < 2n + 1 ≤ 7. -5 < 2n ≤ 6 ⇒ -2.5 < n ≤ 3, so n = -2, -1, 0, 1, 2, 3.
🔍 Worked example — Easy
Find all integer values of n that satisfy the compound inequality: -4 < 2n + 1 ≤ 7.
🔍 Worked example — Standard
A region is defined by the inequalities y ≥ 1, x + y ≤ 6, and y < 2x. Describe the boundary lines and determine if the point (2, 2) lies within the region.
🔍 Worked example — Exam-style
A region is bounded by the lines x = 1 (solid), y = 4 (broken), and y = x (solid). The region contains the point (2, 3). List the inequalities that define this region.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 2x ≥ 10 ⇒ x ≥ 5. (2) -5x < 15 ⇒ x > -3. (3) -2 ≤ x < 4: -2, -1, 0, 1, 2, 3. (4) x ≥ 0, y ≥ 0, 2x + y ≤ 8.