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Linear Inequalities And Shading

Linear Inequalities And Shading

Inequalities on graphs: regions and shading

Shading Rules Extended 8 min read

On this page
  1. In this note
  2. Formula bank
  3. One line, two sides
  4. Writing the inequalities for a region
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.6: inequalities on graphs
  • 2.6Show an inequality as a region: solid line for \(\le,\ \ge\), broken line for \(<,\ >\).
  • 2.6Find and label a region R from several inequalities, shading the unwanted regions.
  • 2.6Write down the inequalities that define a region drawn on a grid.

Before you start: Solving inequalities and straight lines (equation of a line). Next: Inequality problems.

Formula bank

Line styleLearn it\(\le,\ \ge\): solid line; \(<,\ >\): broken linesolid = points on the line are included.Use it when drawing any boundary.
Test pointLearn itput a point not on the line into the inequalitytrue → that side is wanted; false → the other side.Use it when deciding which side to shade.

One line, two sides

The line \(x+y=6\) is every point whose coordinates add to exactly 6. On one side \(x+y<6\), on the other \(x+y>6\). So an inequality is a whole half of the grid, and a test point tells you which half.

General methodHow to find and label a region R

  1. Draw each boundary line (change \(\le\) to \(=\) to get its equation): solid for \(\le,\ \ge\), broken for \(<,\ >\).
  2. Test a point that is NOT on the line - (0, 0) is easiest. True: its side is wanted. False: the other side is.
  3. Shade the side you do NOT want (the usual instruction), for every line.
  4. The clear part left is R. Label it R.
Region R for three inequalities-11234567-11234567Rxy
Unwanted regions shaded; R is left clear.
  1. \(y\ge1\): test (0, 0): \(0\ge1\) false → wanted side is above \(y=1\).
  2. \(x+y\le6\): \(0\le6\) true → wanted side contains (0, 0).
  3. \(y\le2x\): (0, 0) is ON the line, so test (1, 0): \(0\le2\) true.
  4. All solid: points on the edges are in R.

A strict inequality such as \(y>x-1\) uses a broken line: a point exactly on it, like (3, 2), is NOT in R.

Try it yourself

1
Is (0, 0) in the region \(y\ge x+1\)? Type yes or no.
Hint
\(0\ge1\)?
Show answer

Answer no

2
How is the boundary of \(y>3x-2\) drawn? Type solid or broken.
Show answer

Answer broken

Writing the inequalities for a region

General methodHow to write the inequalities that define R

  1. Find the equation of each boundary line (vertical: \(x=a\); horizontal: \(y=b\); sloping: \(y=mx+c\)).
  2. Pick a point inside R and test it in each equation to see which way round the inequality goes.
  3. Solid line → \(\le\) or \(\ge\); broken line → \(<\) or \(>\).

Example R is bounded by the axes and the line through (0, 6) and (3, 0), and contains (1, 1). Write the three inequalities.

Equation of the sloping line, then test (1, 1).

  1. gradient \(=-6\div3=-2\), intercept 6: \(y=-2x+6\), i.e. \(2x+y=6\)
  2. (1, 1): \(2+1=3\le6\)
  3. also \(x\ge0\), \(y\ge0\)

Answer \(x\ge0,\ y\ge0,\ 2x+y\le6\)

Region bounded by axes and a line-112345-11234567Rxy

Try it yourself

1
R is bounded by \(x=1\), \(y=1\) and \(x+y=5\) (solid) and contains (2, 2). Give the inequality for the line \(x+y=5\).
Show answer

Answer \(x+y\le5\)

Ways they test it

Type 1 Find and label R

"By drawing suitable lines and shading the unwanted regions, find and label the region R that satisfies ..." (3-5 marks)

Spot it a list of inequalities and a blank grid.

  1. Draw each line with the right style.
  2. Test points; shade the unwanted sides.
  3. Label R.

Type 2 Write the inequalities

"Find the three inequalities that define the region R." (3-4 marks)

Spot it a shaded diagram is given.

  1. Equation of each line.
  2. Test a point inside.
  3. Solid/broken decides ≤ or <.

Type 3 Points in R

"How many points with integer coordinates lie in R?" or "the point in R with the largest ..."

Spot it after R is drawn.

  1. Go column by column.
  2. Edges count only if the lines are solid.

For example Points in the region in the first diagram

  1. x = 1: 2; x = 2: 4; x = 3: 3; x = 4: 2; x = 5: 1

Answer 12

Mistakes to avoid

  • ✗ Solid line for <✓ broken lineThe boundary is not included.
  • ✗ Shading the wanted region✓ shade the unwanted (if asked)Follow the instruction.
  • ✗ Testing (0, 0) when it is on the line✓ use another pointIt tells you nothing.
  • ✗ R not labelled✓ write R in the regionThat mark is easy to lose.

Summary

  • ≤, ≥: solid line. <, >: broken line.
  • Test a point off the line to choose the side.
  • Shade the unwanted side of every line; R is what's left - label it.
  • To describe R: equation of each line, then test an inside point.

If you see …

  • "shade the unwanted regions"shade OUTSIDE R.
  • a broken boundaryuse < or >, never ≤ or ≥.

Practice

Practice: easy, medium, hard

Easy

1
Is (0, 0) in the region \(x+y\le6\)? Type yes or no.
Show answer

Answer yes

2
A region lies above the solid line \(y=1\). Write its inequality.
Show answer

Answer \(y\ge1\)

3
A region lies to the left of the broken line \(x=4\). Write its inequality.
Show answer

Answer \(x<4\)

Medium

1
R is bounded by the solid line \(x+2y=8\) and lies on the same side as the origin. Write the inequality.
Guide me

What is it asking? an inequality

What do we know? solid line

Plan (0, 0): \(0\le8\)

Show working
  1. test (0, 0): \(0\le8\) true
  2. \(x+2y\le8\)

Answer \(x+2y\le8\)

2
R is bounded by the axes and the line through (0, 6) and (3, 0), and contains (1, 1). Write the inequality for the sloping line.
Guide me

What is it asking? an inequality

What do we know? gradient −2, intercept 6

Plan (1, 1): \(3\le6\)

Show working
  1. \(y=-2x+6\Rightarrow2x+y=6\)
  2. \(2x+y\le6\)

Answer \(2x+y\le6\)

3
How many points with integer coordinates satisfy \(x\ge1\), \(y\ge1\), \(x+y\le5\)?
Guide me

What is it asking? a count

What do we know? go column by column

Plan x = 1: y = 1 to 4, …

Show working
  1. x = 1: 4; x = 2: 3; x = 3: 2; x = 4: 1
  2. total 10

Answer 10

Hard

1
How many points with integer coordinates satisfy \(y\ge1\), \(x+y\le6\), \(y\le2x\)?
Guide me

What is it asking? a count

What do we know? draw or list column by column

Plan edges count (all solid)

Show working
  1. x = 1: y = 1, 2
  2. x = 2: y = 1 to 4
  3. x = 3: 3; x = 4: 2; x = 5: 1
  4. 2 + 4 + 3 + 2 + 1 = 12

Answer 12

PartAnswer / workingMarksGuidance
correct regionM1
12A1
Total: 2 marks
2
How many points with integer coordinates satisfy \(x\ge0\), \(y\ge0\), \(2x+y\le6\)?
Guide me

What is it asking? a count

What do we know? x = 0, 1, 2, 3

Plan y from 0 up to 6 − 2x

Show working
  1. x = 0: 7; x = 1: 5; x = 2: 3; x = 3: 1
  2. total 16

Answer 16

PartAnswer / workingMarksGuidance
correct methodM1
16A1
Total: 2 marks
3
Is the point (3, 2) in the region \(y>x-1\), \(y\le4\), \(x\ge0\)? Type yes or no.
Guide me

What is it asking? yes or no

What do we know? \(y>x-1\) is strict

Plan (3, 2) is ON the line \(y=x-1\)

Show working
  1. \(2>3-1\) is false (2 = 2)
  2. so (3, 2) is not in R

Answer no

PartAnswer / workingMarksGuidance
no, with reasonB1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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