GCSE: Inequalities – Shading Regions

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified: 31st August 2015

Associated Resources: GCSE-InequalitiesRegionsWorksheet.docx

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Means: x is less than or equal to 4.

x > 3Means: x is (strictly) greater than 3.

0 1 2 3 4 5

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This is known as a ‘strict’ inequality.

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x < -1Means: x is (strictly) less than -1.

-3 -2 -1 0 1 2

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x ≥ 4Means: x is greater than or equal to 4.

2 3 4 5 6 7

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x ≤ 5

2 3 4 5 6 7

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RECAP: Inequalities on a Number Line

STARTER: Solving Inequalities

−𝟐 ,−𝟏 ,𝟎 ,𝟏 ,𝟐 ,𝟑𝒙<𝟒

𝒙<−𝟒

𝟔<𝒙 ≤𝟗

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(On your worksheet)

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

Sketch the region

representing

Think: How are the values of x constrained, and how are the values of y constrained?

can be any number greater than 2, but can be ANY value because it is not constrained.

The dotted line indicates that can’t be 2.

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

-4 < y ≤ -2

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

y ≥ x

How can we tell which way round the inequality is?Just try a point on one side, e.g. (0, 4). Is y ≥ x? Yes, so we’re on the right side.

Quickfire RegionsIs the region on the correct side of the line?

y > x + 2

2

WinFail

y > 1 - x1

WinFail

y – x > 1

1

WinFail

x + y > 11

WinFail

x > 2y - 3

1.5

y > -x

WinFail

Bro Tip: An easier way to see if the region is on the correct side is to see whether is on the smaller side of the inequality or the greater side. If the greater, we’re above.

WinFail

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

y ≤ x + 1 and x ≤ 6 and y > 2

Combining Inequalities

Bro Tip: In GCSE exam questions, the region you want is almost always the region enclosed by your 3 lines.

Question 4 Worksheet Answers

Click to Sketch >

Question 5 Worksheet Answers

Click to Sketch >

Question 6 Worksheet Answers

(𝟐 ,𝟐 ) , (𝟐 ,𝟑 ) , (𝟐 ,𝟒 ) ,(𝟑 ,𝟑)?

Click to Sketch > R

Question 7 Worksheet Answers

Click to Sketch >

Question 8 Worksheet Answers

3 2

Click to Sketch >

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Question 9 Worksheet Answers

𝟓𝟐−

𝟓𝟔 𝒙

𝟐𝟎

R

1 1

Click to Sketch >

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