Quick revision notes · Mathematics 4024 · Chapter 17 of 35
Linear Inequalities And Shading
Inequality symbols, number lines, solving inequalities, and regions on a graph.
3 sectionsWorked examples
Common mistakes
At a glance
Inequality symbols, number lines, solving inequalities, and regions on a graph.
1Symbols and number lines
- < less than, > greater than, ⩽ less than or equal to, ⩾ greater than or equal to.
- On a number line: open circle for < or >; filled circle for ⩽ or ⩾.
- Double inequality −2 < x ⩽ 3 is drawn between the two values: open at −2, filled at 3.
| Inequality | Circle | Arrow |
|---|---|---|
| x > 2 | open at 2 | right |
| x ⩾ 2 | filled at 2 | right |
| x < −1 | open at −1 | left |
| −2 < x ⩽ 3 | open at −2, filled at 3 | between |
Worked example
List the integers n with −2 < n ⩽ 3.
- −2 is not included; 3 is.
−1, 0, 1, 2, 3
2Solving inequalities
- Solve like an equation: add, subtract, multiply or divide both sides.
- Multiplying or dividing by a negative number reverses the sign.
- Double inequalities: do the same to all three parts.
Worked example
Solve −3 ⩽ 2x + 1 < 7 and list the integer solutions.
- Subtract 1: −4 ⩽ 2x < 6.
- Divide by 2: −2 ⩽ x < 3.
x = −2, −1, 0, 1, 2
Exam tip: To avoid flipping, move the x term to the side where it is positive: 4 − 2x ⩽ 10 → −6 ⩽ 2x → x ⩾ −3.
3Regions on a graph
- Draw each boundary line: solid for ⩽ or ⩾, broken for < or >.
- Pick a test point not on the line (usually (0, 0)) to decide which side you want.
- Read the question: shade the region you want, or the region you don't want (unwanted region).
- y ⩾ 1 is above the line y = 1; x ⩽ 3 is to the left of x = 3.
How to do it
- Draw the line (make a table or use y = mx + c).
- Decide solid or broken.
- Test a point: does it make the inequality true?
- Shade the correct side.
Worked example
The region R satisfies x ⩾ 1, y ⩾ 2 and x + y ⩽ 8. Find the greatest value of 3x + 2y for integer points in R.
- Corners of R: (1, 2), (6, 2), (1, 7).
- 3x + 2y: 7, 22 and 17.
22, at (6, 2)
Exam tip: Maximum and minimum values of an expression in a region are always at or near a corner.
✗Common mistakes
| Don’t write | Write instead |
|---|---|
| −2x ⩽ 6 → x ⩽ −3 | Dividing by −2 flips the sign: x ⩾ −3 |
| Including −2 in −2 < n | < means −2 is not included |
| Drawing y < x + 3 with a solid line | Strict inequality → broken line |
| Shading the wanted region when the question says shade the unwanted one | Read which region to shade |
★In the exam
- Label every boundary line with its equation.
- Use (0, 0) as a test point when it is not on the line — the arithmetic is easy.
- When listing integers, check the end values against the symbols.