DP IB Maths: AA HL

Revision Notes

IBMaths: AA HLDPRevision Notes2. Functions2.9 Further Functions & Graphs2.9.3 Modulus Equations & Inequalities

2.9.3 Modulus Equations & Inequalities

Test Yourself

Modulus Equations

How do I find the modulus of a function?

  • The modulus of a function f(x) is
    • open vertical bar f left parenthesis x right parenthesis close vertical bar equals open curly brackets table row cell f left parenthesis x right parenthesis end cell row cell negative f left parenthesis x right parenthesis end cell end table blank table row cell f left parenthesis x right parenthesis greater or equal than 0 end cell row cell f left parenthesis x right parenthesis less than 0 end cell end table closeor
    • open vertical bar f left parenthesis x right parenthesis close vertical bar equals square root of left square bracket f left parenthesis x right parenthesis right square bracket squared end root

How do I solve modulus equations graphically?

  • To solve |f(x)| = g(x) graphically
    • Draw y = |f(x)| and y = g(x) into your GDC
    • Find the x-coordinates of the points of intersection

How do I solve modulus equations analytically?

  • To solve |f(x)| = g(x) analytically
    • Form two equations
      • f(x) = g(x)
      • f(x) = - g(x)
    • Solve both equations
    • Check solutions work in the original equation
      • For example: x minus 2 equals 2 x minus 3 has solution x equals 1
      • But vertical line left parenthesis 1 right parenthesis minus 2 vertical line equals 1 and 2 left parenthesis 1 right parenthesis minus 3 equals negative 1
      • So x equals 1 is not a solution to vertical line x minus 2 vertical line equals 2 x minus 3

Worked example

Solve for x:

a)
stretchy vertical line fraction numerator 2 x plus 3 over denominator 2 minus x end fraction stretchy vertical line equals 5

2-8-3-ib-aa-hl-modulus-equation-a-we-solution

b)
open vertical bar 3 x minus 1 close vertical bar equals 5 x minus 11.

2-8-3-ib-aa-hl-modulus-equation-b-we-solution

Modulus Inequalities

How do I solve modulus inequalities analytically?

  • To solve any modulus inequality
    • First solve the corresponding modulus equation
      • Remembering to check whether solutions are valid
    • Then use a graphical method or a sign table to find the intervals that satisfy the inequality
  • Another method is to solve two pairs of inequalities
    • For |f(x)| < g(x) solve:
      • f(x) < g(x) when f(x) ≥ 0
      • f(x) > -g(x) when f(x) ≤ 0
    • For |f(x)| > g(x) solve:
      • f(x) > g(x) when f(x) ≥ 0
      • f(x) < -g(x) when f(x) ≤ 0

Exam Tip

  • If a question on this appears on a calculator paper then use the same ideas as solving other inequalities
    • Sketch the graphs and find the intersections

Worked example

Solve the following inequalities for x.

a)
vertical line 2 x minus 1 vertical line less than 4

2-8-3-ib-aa-hl-modulus-inequality-a-we-solution

b)
open vertical bar x plus 1 close vertical bar less than open vertical bar 2 x plus 3 close vertical bar

K-a4iR1J_2-8-3-ib-aa-hl-modulus-inequality-b-we-solution

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    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.