IBMaths: AA HLDPRevision Notes2. Functions2.8 Inequalities2.8.1 Solving Inequalities Graphically

Solving Inequalities Graphically (DP IB Maths: AA HL)

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Solving Inequalities Graphically

How can I solve inequalities graphically?

  • Consider the inequality f(x) ≤ g(x), where f(x) and g(x) are functions of x
    • if we move g(x) to the LHS we get
      • f(x) – g(x) ≤ 0
  • Solve f(x) – g(x) = 0 to find the zeros of f(x) – g(x)
    • These correspond to the x-coordinates of the points of intersection of the graphs y = f(x) and y = g(x)
  • To solve the inequality we can use a graph
    • Graph y = f(x) – g(x) and label its zeros
    • Hence find the intervals of x that satisfy the inequality f(x) – g(x) ≤ 0
      • These are the intervals which satisfies the original inequality f(x) ≤ g(x)
    • This method is particularly useful when finding the intersections between the functions is difficult due to needing large x and y windows on your GDC

Be careful when rearranging inequalities!

  • Remember to flip the sign of the inequality when you multiply or divide both sides by a negative number
    • e. 1 < 2 → [times both sides by (–1)] → –1 > –2 (sign flips)
  • Never multiply or divide by a variable as this could be positive or negative
    • You can only multiply by a term if you are certain it is always positive (or always negative)
      • Such as x squared comma space open vertical bar x close vertical bar comma space straight e to the power of x
  • Some functions reverse the inequality
    • Taking reciprocals of positive values
      • 0 less than x less than y rightwards double arrow 1 over x greater than 1 over y
    • Taking logarithms when the base is 0 < a < 1
      • 0 less than x less than y rightwards double arrow log subscript a open parentheses x close parentheses greater than log subscript a open parentheses y close parentheses
  • The safest way to rearrange is simply to add & subtract to move all the terms onto one side

Worked example

Use a GDC to solve the inequality 2 x cubed less than x to the power of 5 minus 2 x.

2-8-1-ib-aa-hl-graphical-inequalities-we-solution

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    Lucy

    Author: Lucy

    Expertise: Head of STEM

    Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.