A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 3: Equations and inequalities · Lesson 3

Inequalities on graphs and regions

70 marks

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Date

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Quick questions to start.

A1
Needed today
Find the point where the lines $y = 3 x - 4$ and $x + y = 8$ meet.
[2 marks]
Answer $(\dots\dots, \dots\dots)$
A2
Last lesson
Find the set of values of $x$ for which $x^{2} - 2 x - 24 > 0$ .
[2 marks]
Answer
A3
From chapter 2
Work out the discriminant of $2 x^{2} - 5 x + 4$ . How many real roots does $2 x^{2} - 5 x + 4 = 0$ have?
[2 marks]
Answer
A4
From chapter 1
Write $98 + 18$ in the form $k 2$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Use the sketch to solve $\frac{6}{x} < x - 1$ , $x \neq = 0$ , giving your answer in set notation.

Hint Find where the graphs meet. The curve has two branches, so

x = 0

is a critical value too.

B1
By sketching $y = \frac{8}{x}$ and $y = x + 2$ on the same axes, find, in set notation, the set of values of $x$ for which $\frac{8}{x} > x + 2$ , $x \neq = 0$ .
[4 marks]
Answer

Example 2

Draw on a grid the region $R$ that satisfies $y > - 1$ , $y \leq x + 2$ and $x + y < 6$ . Find the coordinates of the vertices of $R$ and the area of $R$ . Which vertices belong to $R$ ?

Hint A strict inequality (

<

>

) has a dashed line;

\leq

\geq

has a solid line. A vertex belongs to

R

only if both lines through it are solid.

B2
The region $R$ is the set of points with $y \geq - 1$ , $y < 2 x + 1$ and $2 x + y \leq 9$ . Find the vertices of $R$ and the area of $R$ .
[4 marks]
Answer

Example 3

The sketch shows the curve $y = 4 x - x^{2}$ and the line $y = x$ . The line is dashed and the curve is solid. Write down the inequalities that define the shaded region $R$ .

Hint Is

R

above or below each boundary? A solid boundary is part of

R

, so use

\leq

\geq

for it.

B3
The sketch shows the curve $y = x^{2} - 6 x + 5$ and the line $y = x - 1$ . The line is dashed; the curve and the $x$ -axis are solid. Write down the inequalities that define the shaded region $R$ .
[3 marks]
Answer

Practice

A-Level

C1
On the grid, shade the region $R$ that satisfies $x \geq 0$ , $y \geq x - 2$ and $x + y < 4$ . Find the area of $R$ , and state which vertices of $R$ belong to $R$ .
[5 marks]
Answer

C2
The region $R$ is the set of points with $y \geq x^{2} - 2 x - 3$ and $y < x + 1$ . Which of the points $P (0, 0)$ , $Q (4, 5)$ and $S (- 1, 0)$ lie in $R$ ? Give a reason for each.
[3 marks]
Answer
C3
The sketch shows the lines $x = - 2$ , $y = 2 x + 2$ and $y = 8 - x$ . The line $y = 8 - x$ is dashed; the other two are solid. Write down the inequalities that define the shaded region $R$ , and find the area of $R$ .
[4 marks]
Answer
C4
Find the set of values of $k$ for which the curve $y = x^{2} + k x + 9$ lies entirely above the line $y = x$ .
[4 marks]
Answer

Exam-style questions

Edexcel exam style

D1
The curve $C$ has equation $y = 2 x^{2} - 5 x - 3$ and the line $l$ has equation $y = x + 5$ . The line meets the curve at the points $A$ and $B$ , as shown. The region $R$ , shown shaded, is bounded by $C$ , $l$ and the $x$ -axis, and includes its boundaries.
(a)
Find the coordinates of $A$ and of $B$ .
[4 marks]
Answer $A (\dots\dots, \dots\dots)$ $B (\dots\dots, \dots\dots)$
(b)
Use inequalities to define the region $R$ .
[3 marks]
Answer
(c)
Hence, or otherwise, find the set of values of $x$ for which $2 x^{2} - 5 x - 3 > x + 5$ .
[2 marks]
Answer
Total for D1: 9 marks

D2
Priya solves $\frac{10}{x} > x - 3$ , $x \neq = 0$ . She writes: "I multiply both sides by $x$ to get $10 > x^{2} - 3 x$ . So $x^{2} - 3 x - 10 < 0$ , $(x - 5) (x + 2) < 0$ , and the answer is $- 2 < x < 5$ ."
Priya is wrong.
(a)
Explain the mistake in Priya's method, and give a value of $x$ in her answer that does not satisfy $\frac{10}{x} > x - 3$ .
[2 marks]
(b)
Find, in set notation, the correct set of values of $x$ for which $\frac{10}{x} > x - 3$ .
[4 marks]
Answer
Total for D2: 6 marks
D3
The curve $C$ has equation $y = x^{2} + 4$ . The line $l$ has equation $y = m x$ , where $m$ is a positive constant. The line $l$ is a tangent to $C$ at the point $P$ .
(a)
Find the value of $m$ .
[3 marks]
Answer $m = \dots\dots$
(b)
Hence find the coordinates of $P$ .
[2 marks]
Answer $P (\dots\dots, \dots\dots)$
Total for D3: 5 marks
D4
The shaded region $R$ in the sketch is the set of points with $y \leq 3 - 2 x - x^{2}$ and $y \geq x^{2} + 4 x - 5$ .
(a)
Find the coordinates of the points where the two curves meet.
[4 marks]
Answer
(b)
Hence find, in set notation, the set of values of $x$ for which $3 - 2 x - x^{2} \geq x^{2} + 4 x - 5$ .
[2 marks]
Answer
(c)
Find the greatest value and the least value of $y$ for points in $R$ .
[3 marks]
Answergreatest $\dots\dots$ least $\dots\dots$
Total for D4: 9 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
How many points with integer coordinates $(x, y)$ satisfy both $y \geq x^{2} - 9$ and $y < 9 - x^{2}$ ?
[6 marks]
Hint 1 · What to try
Take one value of $x$ at a time. For which whole numbers $x$ can there be any $y$ at all?
Hint 2 · The first line
For a fixed whole number $x$ , $y$ runs from $x^{2} - 9$ up to $8 - x^{2}$ , so there are $(8 - x^{2}) - (x^{2} - 9) + 1 = 18 - 2 x^{2}$ values.
Hint 3 · The full method
$18 - 2 x^{2} > 0$ needs $x^{2} < 9$ , so $x = - 2, - 1, 0, 1, 2$ . That gives $10 + 16 + 18 + 16 + 10 = 70$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can read the solution of an inequality from two graphs.
I can handle a reciprocal graph, where x = 0 is a critical value.
I can shade a region using dashed and solid boundaries.
I can write down the inequalities that define a shaded region.

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