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Solving Inequalities (AQA GCSE Maths)

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  • FrontSolving Linear Inequalities

    Define the word inequality in algebra.

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  • Define the word inequality in algebra.

    An inequality compares a left-hand side to a right-hand side and states which one is bigger, using the symbols less than comma space greater than comma space less or equal than comma space greater or equal than.

  • Explain the meaning of the word linear in linear inequality.

    The word linear in linear inequality means that the terms in the inequality are either constant numbers or terms in x, but not terms in x squared or x cubed etc.

    These are examples of linear inequalities:

    • x plus 2 greater than 5

    • 2 x less than x minus 1

  • True or False?

    You can add or subtract terms to both sides of a linear inequality in exactly the same way as you do to a linear equation.

    True.

    You can add or subtract terms to both sides of a linear inequality in exactly the same way as you do to a linear equation.

  • True or False?

    You can multiply or divide both sides of a linear inequality in exactly the same way as you do to a linear equation.

    False.

    You can multiply or divide both sides of a linear inequality in exactly the same way as you do to a linear equation as long as you multiply or divide by positive numbers.

    If, however, you multiply or divide both sides by negative numbers, you have to flip the direction of the inequality sign.

    E.g. you can divide table row cell 2 x end cell less than 4 end table by 2 to get table row x less than 2 end table.
    You can divide table row cell negative 2 x end cell less than 4 end table by -2, but you must flip the inequality to get table row x greater than cell negative 2 end cell end table.

  • How do number lines highlight the difference between strict inequalities (such as x less than 3) and non-strict inequalities (such as x less or equal than 3)?

    Number lines show an open circle for strict inequalities, e.g. x less than 3, and a closed circle for non-strict inequalities, e.g. x less or equal than 3.

    Two number lines labelled from -5 to +5. The top number line shows an arrow from 3, marked by an open circle, to negative direction. It is labelled 'x < 3'. The bottom number line shows an arrow from 3, marked by a closed circle, to negative direction. It is labelled 'x ≤ 3'.

  • True or False?

    The number line representing " x less than 1 or x greater than 3" consists of two separate arrows pointing outwards in opposite directions.

    True.

    The number line representing " x less than 1 or x greater than 3" consists of two separate arrows pointing outwards in opposite directions.

    A number line from -5 to 5. An empty circle is at 1 with an arrow pointing in the negative direction. Another empty circle is at 3 with an arrow pointing in the positive direction.

  • True or False?

    The diagram below shows the inequality negative 4 less than x less or equal than 2.

    A number line from -5 to 5. Empty circles are located at -4 and 2 with a horizontal line joining them.

    False.

    The number line in the diagram does not show the inequality negative 4 less than x less or equal than 2.

    It has two open circles which indicate the inequality negative 4 less than x less than 2.

  • Explain how you would solve an inequality in the form negative 10 less than a x plus b less than 10.

    To solve an inequality in the form negative 10 less than a x plus b less than 10,

    1. Subtract b from all three parts.

    2. Then divide all three parts by a.

    E.g. negative 10 less than 2 x plus 6 less than 10 minus 16 less than 2 x less than 16 minus 8 less than x less than 8

    (Though if a is negative, remember to 'flip' the inequality signs when dividing.)

    An alternative method is to split into two different inequalities, negative 10 less than 2 x plus 6 and 2 x plus 6 less than 10, then solve these individually.

  • How would "x is greater than or equal to 3" be written using set notation?

    "x is greater than or equal to 3" would be written using set notation as:

    open curly brackets x colon space x greater or equal than 3 close curly brackets

  • How would "x is greater than 3 and less than 6" be written using set notation?

    "x is greater than 3 and less than 6" would be written using set notation as:

    open curly brackets x colon space x greater than 3 close curly brackets space intersection space open curly brackets x colon space x less than 6 close curly brackets or open curly brackets x colon space 3 less than x less than 6 close curly brackets

  • How would "x is less than 2 or greater than 8" be written using set notation?

    "x is less than 2 or greater than 8" would be written using set notation as:

    open curly brackets x colon space x less than 2 close curly brackets space union space open curly brackets x colon space x greater than 8 close curly brackets

  • Outline how to solve a quadratic inequality, such as a x squared plus b x plus c greater than 0.

    To solve a quadratic inequality such as a x squared plus b x plus c greater than 0:

    • Find the roots of the quadratic equation a x squared plus b x plus c equals 0.

    • Sketch a graph of the quadratic and label the roots.

      • If it is a positive quadratic it will be U-shaped.

      • If it is a negative quadratic it will be n-shaped.

    • Identify the region that satisfies the inequality.

      • For a x squared plus b x plus c greater than 0 the region above the x-axis satisfies the inequality.

  • The graph of y equals x squared minus 7 x plus 10 is shown below.

    Use the graph to find the answer to x squared minus 7 x plus 10 greater than 0.

    Graph of a positive quadratic intersecting the x-axis at points 2 and 5

    Shade the region which satisfies x squared minus 7 x plus 10 greater than 0 (above the x-axis).

    The answer is x greater than 5 or x less than 2.

    Positive quadratic, intersecting the x-axis at points 2 and 5, with the area between the x axis and the curve shaded in red.

  • The graph of y equals x squared minus 2 x minus 15 is shown below.

    Use the graph to find the answer to x squared minus 2 x minus 15 less than 0.

    Graph of a positive quadratic intersecting the x-axis at points -3 and 5

    Shade the region which satisfies x squared minus 2 x minus 15 less than 0 (below the x-axis).

    The answer is negative 3 less than x less than 5.

    Positive quadratic, intersecting the x-axis at points -3 and 5, with the area between the x axis and the curve shaded in red.

  • The graph of y equals negative x squared plus 4 x plus 12 is shown below.

    Use the graph to find the answer to negative x squared plus 4 x plus 12 greater than 0.

    Graph of a negative quadratic intersecting the x-axis at points -2 and 6

    Shade the region which satisfies negative x squared plus 4 x plus 12 greater than 0 (above the x-axis).

    The answer is negative 2 less than x less than 6.

    Negative quadraic, intersecting the x-axis at points -2 and 6, with the area between the x axis and the curve shaded in red.
Previous:Real-Life GraphsNext:Graphing Inequalities

    1. Topic 1

      Number Operations

    2. Topic 2

      Prime Factors, HCF & LCM

    3. Topic 3

      Powers, Roots & Standard Form

    4. Topic 4

      Fractions

    5. Topic 5

      Percentages

    6. Topic 6

      Simple & Compound Interest, Growth & Decay

    7. Topic 7

      Fractions, Decimals & Percentages

    8. Topic 8

      Rounding, Estimation & Bounds

    9. Topic 9

      Surds

    10. Topic 10

      Using a Calculator

    1. Topic 1

      Introduction to Algebra

    2. Topic 2

      Algebraic Roots & Indices

    3. Topic 3

      Expanding Brackets

    4. Topic 4

      Factorising

    5. Topic 5

      Completing the Square

    6. Topic 6

      Algebraic Fractions

    7. Topic 7

      Rearranging Formulas

    8. Topic 8

      Algebraic Proof

    9. Topic 9

      Linear Equations

    10. Topic 10

      Quadratic Equations

    11. Topic 11

      Simultaneous Equations

    12. Topic 12

      Iteration

    13. Topic 13

      Forming & Solving Equations

    14. Topic 14

      Functions

    15. Topic 15

      Coordinate Geometry

    16. Topic 16

      Linear Graphs

    17. Topic 17

      Graphs of Functions

    18. Topic 18

      Equation of a Circle

    19. Topic 19

      Estimating Gradients & Areas under Graphs

    20. Topic 20

      Real-Life Graphs

    21. Topic 21

      Solving Inequalities

    22. Topic 22

      Graphing Inequalities

    23. Topic 23

      Transformations of Graphs

    24. Topic 24

      Sequences

    1. Topic 1

      Ratios

    2. Topic 2

      Problem Solving with Ratios

    3. Topic 3

      Direct & Inverse Proportion

    4. Topic 4

      Standard & Compound Units

    5. Topic 5

      Exchange Rates & Best Buys

    1. Topic 1

      Symmetry & Shapes

    2. Topic 2

      Angles in Polygons & Parallel Lines

    3. Topic 3

      Bearings, Scale Drawing, Constructions & Loci

    4. Topic 4

      Circle Theorems

    5. Topic 5

      Area & Perimeter

    6. Topic 6

      Circles, Arcs & Sectors

    7. Topic 7

      Volume & Surface Area

    8. Topic 8

      Congruence, Similarity & Geometrical Proof

    9. Topic 9

      Area & Volume of Similar Shapes

    10. Topic 10

      Pythagoras & Trigonometry

    11. Topic 11

      Sine, Cosine Rule & Area of Triangles

    12. Topic 12

      3D Pythagoras & Trigonometry

    13. Topic 13

      Vectors

    14. Topic 14

      Transformations

    1. Topic 1

      Introduction to Probability

    2. Topic 2

      Simple Probability Diagrams

    3. Topic 3

      Tree Diagrams

    4. Topic 4

      Combined & Conditional Probability

    1. Topic 1

      Averages, Ranges & Data

    2. Topic 2

      Statistical Diagrams

    3. Topic 3

      Histograms

    4. Topic 4

      Cumulative Frequency & Box Plots

    5. Topic 5

      Scatter Graphs & Correlation