IGCSE · Practice questions · Inequalities as regions on a graph

Fence it in.

Extended. Six original Cambridge-style questions on inequalities in two variables: boundary lines, choosing the side, defining a region and listing integer points.

Original questions All questions on this page are original work, written in the Cambridge IGCSE style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Line, type, side, shade.

Line: change the sign to \(=\) and draw the line (use intercepts or \(y = mx + c\)).
Type: solid for \(\le, \ge\); broken for \(<, >\).
Side: test a point not on the line, usually \((0, 0)\).
Shade: the unwanted side, leaving the region R clear.
Integer points: go row by row, and check points on each boundary against its sign.

Recall

[2 marks]

Write down the inequality for (a) the region to the left of the solid line \(x = 3\), (b) the region above the broken line \(y = -2\).

(a) \(x \le 3\) ✓
(b) \(y > -2\) ✓

Recall

[2 marks]

State whether each boundary line is solid or broken. (a) \(y \ge 2x\) (b) \(x + y < 5\)

(a) Solid, because \(\ge\) includes the line. ✓
(b) Broken, because \(<\) excludes the line. ✓

Calculation

[2 marks]

Do the points (a) \((3, -1)\) and (b) \((4, 0)\) satisfy \(2x - y > 7\)?

(a) \(2(3) - (-1) = 7\), and \(7 > 7\) is false. No: the point is on the boundary, which is excluded. ✓
(b) \(2(4) - 0 = 8 > 7\). Yes. ✓

Calculation

[3 marks]

A broken line passes through \((0, 2)\) and \((4, 0)\). The wanted region is the side NOT containing the origin. Find the inequality.

Gradient \(= \dfrac{0 - 2}{4 - 0} = -\dfrac{1}{2}\), so \(y = 2 - \dfrac{1}{2}x\), i.e. \(x + 2y = 4\). ✓
The origin gives \(0 < 4\), so the origin's side is \(x + 2y < 4\). ✓
The wanted side is the other one, and the line is broken: \(x + 2y > 4\). ✓

Analysis

[3 marks]

The region R satisfies \(x \ge 1\), \(y \ge x\) and \(x + y < 6\). List all the points in R with integer coordinates.

\(x = 1\): \(y \ge 1\) and \(y < 5\), so \(y = 1, 2, 3, 4\). ✓
\(x = 2\): \(y \ge 2\) and \(y < 4\), so \(y = 2, 3\). ✓
\(x = 3\): \(y \ge 3\) and \(y < 3\), none. ✓
\((1,1), (1,2), (1,3), (1,4), (2,2), (2,3)\) ✓

Analysis

[5 marks]

A club hires \(x\) small buses and \(y\) large buses. A small bus seats 20 people and a large bus seats 40. At least 120 seats are needed, no more than 5 buses can be hired, and at least one of each type is hired.

(a) Show that \(x + 2y \ge 6\). (b) Write down the other inequalities. (c) List every possible combination \((x, y)\).

(a) Seats: \(20x + 40y \ge 120\). Divide by 20: \(x + 2y \ge 6\). ✓
(b) \(x + y \le 5\), \(x \ge 1\), \(y \ge 1\). ✓
(c) \(y = 1\): \(x \ge 4\) and \(x \le 4\), so \((4, 1)\). ✓
\(y = 2\): \(x \ge 2\) and \(x \le 3\), so \((2, 2), (3, 2)\). ✓
\(y = 3\): \(x \ge 1\) and \(x \le 2\), so \((1, 3), (2, 3)\). \(y = 4\): \(x = 1\), so \((1, 4)\). ✓
Six combinations: \((4,1), (2,2), (3,2), (1,3), (2,3), (1,4)\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.