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Maths Gcse Aqa Higher

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  1. Scatter-Graphs-And-Correlation Aqa Higher
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  2. Cumulative-Frequency-And-Box-Plots Aqa Higher
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  3. Histograms Aqa Higher
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  4. Statistical-Diagrams Aqa Higher
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  5. Averages-Ranges-And-Data Aqa Higher
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  6. Combined-And-Conditional-Probability Aqa Higher
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  7. Tree-Diagrams Aqa Higher
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  8. Simple-Probability-Diagrams Aqa Higher
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  9. Transformations Aqa Higher
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  10. Vectors Aqa Higher
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  11. 3D-Pythagoras-And-Trigonometry Aqa Higher
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  12. Sine-Cosine-Rule-And-Area-Of-Triangles Aqa Higher
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  13. Pythagoras-And-Trigonometry Aqa Higher
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  14. Area-And-Volume-Of-Similar-Shapes Aqa Higher
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  15. Congruence-Similarity-And-Geometrical-Proof Aqa Higher
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  16. Volume-And-Surface-Area Aqa Higher
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  17. Circles-Arcs-And-Sectors Aqa Higher
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  18. Area-And-Perimeter Aqa Higher
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  19. Circle-Theorems Aqa Higher
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  20. Bearings-Scale-Drawing-Constructions-And-Loci Aqa Higher
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  21. Angles-In-Polygons-And-Parallel-Lines Aqa Higher
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  22. Symmetry-And-Shapes Aqa Higher
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  23. Exchange-Rates-And-Best-Buys Aqa Higher
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  24. Standard-And-Compound-Units Aqa Higher
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  25. Direct-And-Inverse-Proportion Aqa Higher
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  26. Problem-Solving-With-Ratios Aqa Higher
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  27. Ratios Aqa Higher
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  28. Sequences Aqa Higher
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  29. Transformations-Of-Graphs Aqa Higher
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  30. Graphing-Inequalities Aqa Higher
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  31. Solving-Inequalities Aqa Higher
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  32. Real-Life-Graphs Aqa Higher
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  33. Estimating-Gradients-And-Areas-Under-Graphs Aqa Higher
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  34. Equation-Of-A-Circle Aqa Higher
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  35. Functions Aqa Higher
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  36. Forming-And-Solving-Equations Aqa Higher
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  37. Graphs-Of-Functions Aqa Higher
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  38. Linear-Graphs Aqa Higher
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  39. Coordinate-Geometry Aqa Higher
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  40. Iteration Aqa Higher
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  41. Simultaneous-Equations Aqa Higher
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  42. Quadratic-Equations Aqa Higher
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  43. Linear-Equations Aqa Higher
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  44. Algebraic-Proof Aqa Higher
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  45. Rearranging-Formulas Aqa Higher
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  46. Algebraic-Fractions Aqa Higher
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  47. Completing-The-Square Aqa Higher
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  48. Factorising Aqa Higher
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  50. Algebraic-Roots-And-Indices Aqa Higher
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  52. Surds Aqa Higher
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  53. Rounding-Estimation-And-Bounds Aqa Higher
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  54. Fractions-Decimals-And-Percentages Aqa Higher
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  55. Introduction Aqa Higher
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  56. Simple-And-Compound-Interest-Growth-And-Decay Aqa Higher
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  57. Percentages Aqa Higher
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  58. Fractions Aqa Higher
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  59. Powers-Roots-And-Standard-Form Aqa Higher
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  61. Number-Operations Aqa Higher
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Exam code:8300

Solving linear inequalities

What is an inequality?

  • An inequality tells you that something is greater than (>) or less than (<) something else

    • x > 5 means x is greater than 5 

      • x could be 6, 7, 8, 9, …

  • Inequalities may also include being equal (=) 

    • ⩾ means greater than or equal to

    • ⩽ means less than or equal to

      • x ⩽ 10 means x is less than or equal to 10

        • x could be 10, 9, 8, 7, 6,….

  • When they cannot be equal, they are called strict inequalities

    • > and < are strict inequalities

      • x > 5 does not include 5 (strict)

      • x ⩾ 5 does include 5 (not strict)

How do I find integers that satisfy inequalities?

  • You may be given two end points and have to list the integer values of x that satisfy the inequality

    • Look at whether each end point is included or not 

      • 3 ⩽ x ⩽ 6

        • x = 3, 4, 5, 6

      • 3 ⩽ x < 6

        • x = 3, 4, 5

      • 3 < x ⩽ 6

        • x = 4, 5, 6

      • 3 < x < 6

        • x = 4, 5

  • If only one end point is given, there are an infinite number of integers

    • x > 2

      • x = 3, 4, 5, 6, …

    • x ⩽ 2

      • x = 2, 1, 0, -1, -2, …

      • Remember zero and negative whole numbers are integers

      • If the question had said positive integers only then just list x = 2, 1

  • You may be asked to find integers that satisfy two inequalities

    • 0 < x < 5 and x ⩾ 3

      • List separately: x = 1, 2, 3, 4 and x = 3, 4, 5, 6, …

      • Find the values that appear in both lists: x = 3, 4 

  • If the question does not say x is an integer, do not assume x is an integer!

    • x > 3 actually means any value greater than 3

      • 3.1 is possible

      • pi = 3.14159… is possible

  • You may be asked to find the smallest or largest integer

    • The smallest integer that satisfies x > 6.5 is 7

Worked Example

List all the integer values of x that satisfy 

negative 4 less or equal than x less than 2

Integer values are whole numbers 
-4 ≤ x shows that x includes -4, so this is the first integer

x = -4

x < 2 shows that x does not include 2
Therefore the last integer is x = 1

x = 1

For the answer, list all the integers from -4 to 1
Remember integers can be zero and negative

bold italic x bold equals bold minus bold 4 bold comma bold space bold minus bold 3 bold comma bold space bold minus bold 2 bold comma bold space bold minus bold 1 bold comma bold space bold 0 bold comma bold space bold 1

How do I represent an inequality on a number line?

  • The inequality -3 < x ≤ 4 is shown on a number line below

<img

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