Choose exam board for KS4 Computer Science (GCSE)
Choose exam board for KS4 English
Choose exam board for KS4 French
Choose exam board for KS4 Geography
Choose exam board for KS4 German
Choose exam board for KS4 History
Choose tier for KS4 Maths
Choose exam board for KS4 Music
Choose exam board for KS4 Physical education (GCSE)
Choose exam board for KS4 Religious education (GCSE)
Choose exam board for KS4 Spanish

      Solving quadratic inequalities in one variable graphically

      Lesson details

      Learning outcome

      I can solve a quadratic inequality graphically.

      Key learning points

      1. A quadratic equation can be represented graphically
      2. The solutions are where the graph cross the x axis
      3. By studying the graph, you can see where the equation is greater than 0
      4. By studying the graph, you can see where the equation is less than 0
      5. The solution set can be represented algebraically or using set notation

      Keywords

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

      • Quadratic - A quadratic is an equation, graph or sequence where the highest exponent of the variable is 2. The general form for a quadratic is ax^2 + bx + c

      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

      Loading...

      Prior knowledge starter quiz

      6 Questions

      Q1.
      When graphed, a quadratic equation forms...

      an upward curve.
      a straight line.
      a wave.
      Correct answer: a parabola.

      Q2.
      Solve $$x^2-64=0$$.

      $$x=64$$
      $$x=8$$
      $$x=-8$$
      Correct answer: $$x=8$$ and $$x=-8$$
      The equation has no solutions.

      Q3.
      Find the roots of $$x^2-3x-10=0$$.

      $$x=3$$
      $$x=-10$$
      Correct answer: $$x=5$$
      $$x=-5$$
      Correct answer: $$x=-2$$

      Q4.
      This could be the sketch of which one of these quadratics?

      An image in a quiz
      Correct answer: $$x^2-10x+21$$
      $$x^2-4x-21$$
      $$x^2+4x-21$$
      $$x^2+10x+21$$
      $$-x^2+10x-21$$

      Q5.
      What is the minimum value of this quadratic?

      An image in a quiz
      Correct answer: $$-4$$
      $$3$$
      $$7$$
      $$0$$
      $$-3$$

      Q6.
      The solution to $$2x^2-14=x^2-3x+14$$ is $$x=4$$ and $$x=$$ .

      Correct Answer: -7, x=-7

      6 Questions

      Q1.
      $$-5<x<5$$ is the __________ the quadratic inequality $$x^2-25<0$$.

      Correct answer: solution to
      graph of
      inequality of
      root of

      Q2.
      This is the curve $$y=x^2-9$$ Use the graph to solve $$x^2-9<0$$.

      An image in a quiz
      $$x<3$$
      Correct answer: $$-3<x<3$$
      $$x<-3$$ or $$x>3$$
      $$x=-3$$ or $$x=3$$
      This inequality has no solutions.

      Q3.
      This is the curve $$y=x^2-9$$. Use the graph to solve $$x^2-9>0$$.

      An image in a quiz
      $$x>3$$
      $$-3<x<3$$
      Correct answer: $$x<-3$$ or $$x>3$$
      $$x>-3$$ or $$x<3$$
      The inequality has no solutions.

      Q4.
      This is the curve $$y=x^2-9$$. Use the graph to solve $$x^2-9<-9$$.

      An image in a quiz
      $$x=0$$
      $$y<-9$$
      $$x<0$$
      $$x<-3$$ or $$x>3$$
      Correct answer: The inequality has no solutions.

      Q5.
      This is the curve $$y=x^2-4x+4$$. The solution to the inequality $$1<x^2-4x+4<4$$ is $$0<x<1$$ or .

      An image in a quiz
      Correct Answer: 3<x<4, 3 < x < 4

      Q6.
      Rearrange and sketch to solve $$100-x^2<64$$.

      $$x<-8$$ or $$x>8$$
      Correct answer: $$x<-6$$ or $$x>6$$
      $$-6<x<6$$
      $$-8<x<8$$
      $$-10<x<10$$

      To help you plan your 11 maths lesson on: Solving quadratic inequalities in one variable graphically, download all teaching resources for free and adapt to suit your pupils' needs...