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      Solving a set of linear inequalities graphically

      Lesson details

      Learning outcome

      I can extend my thinking to multiple linear inequalities with 2 variables and can show this can be solved graphically to find a solution set.

      Key learning points

      1. Multiple linear inequalities in two variables can be graphed on the same axes
      2. By testing points in the various equations, you can determine the region that satisfies all the inequalities
      3. This region is the solution set

      Keywords

      • Inequality - An inequality is used to show that one expression may not be equal to another.

      • Region - A region is an area that graphically represents the solutions to one or more inequalities. Every coordinate pair within the region satisfies the inequalities that define the region.

      Common misconception

      All inequalities are graphed with solid lines.

      When graphed, strict inequalities are indicated with a dashed line. This is important as it visually tells us that values on the line will not satisfy the inequality.

      Teacher tip

      Encourage pupils to identify the region that satisfies multiple inequalities in different ways: graphically, testing a point, algebraically.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      A region is an area that can graphically represent the solutions to one or more...

      expressions.
      equations.
      Correct answer: inequalities.

      Q2.
      Which values satisfy this inequality? $$x\le-7$$

      Correct answer: $$-7$$
      $$-6$$
      Correct answer: $$-7.5$$
      Correct answer: $$-12\;784$$
      $$8$$

      Q3.
      Which inequality is represented by this region?

      An image in a quiz
      $$x>3$$
      $$x<3$$
      Correct answer: $$y>3$$
      $$y<3$$
      $$y\geq3$$

      Q4.
      Which of these coordinate pairs is in the region satisfied by $$y\geq-x$$?

      Correct answer: $$(0,0)$$
      Correct answer: $$(8,8)$$
      $$(-8,-8)$$
      Correct answer: $$(8,-4)$$
      $$(-8,4)$$

      Q5.
      Which inequality is represented by this region?

      An image in a quiz
      $$y\le5-2x$$
      $$y\geq5-2x$$
      $$y<2x-5$$
      $$y>2x-5$$
      Correct answer: $$y\geq2x-5$$

      Q6.
      Which of these coordinate pairs is in the region satisfied by $$y<4x+5$$?

      Correct answer: $$(0,0)$$
      Correct answer: $$(2,2)$$
      Correct answer: $$(2,-2)$$
      $$(-2,2)$$
      $$(-2,-2)$$

      6 Questions

      Q1.
      This region satisfies both inequalities $$x>-5$$ and $$y<3$$...

      An image in a quiz
      Correct answer: simultaneously.
      separately.
      singularly.

      Q2.
      Which region satisfies both inequalities simultaneously? $$y<-x$$ and $$x>-5$$

      An image in a quiz
      R1
      Correct answer: R2
      R3
      R4

      Q3.
      Which coordinate pairs satisfy both of these inequalities simultaneously? $$y<x$$ and $$y>2$$

      $$(4,4)$$
      $$(4,5)$$
      Correct answer: $$(5,4)$$
      Correct answer: $$(4,3)$$
      $$(3,4)$$

      Q4.
      In which region are all three of these inequalities satisfied simultaneously? $$y<4$$, $$y>x-3$$ and $$y>-x-2$$

      An image in a quiz
      R1
      R2
      R3
      R4
      Correct answer: R5

      Q5.
      In which region are all three of these inequalities satisfied simultaneously? $$y<4$$, $$y<x-3$$ and $$y>-x-2$$

      An image in a quiz
      R1
      Correct answer: R2
      R3
      R4
      R5

      Q6.
      In which region are all three of these inequalities satisfied simultaneously? $$y>4$$, $$y<x-3$$ and $$y<-x-2$$

      An image in a quiz
      R1
      R2
      R3
      R4
      Correct answer: No region can satisfy all three simultaneously

      To help you plan your 11 maths lesson on: Solving a set of linear inequalities graphically, download all teaching resources for free and adapt to suit your pupils' needs...