Modulus Equations & Inequalities (DP IB Analysis & Approaches (AA)): Revision Note

Modulus Equations

How do I find the modulus of a function?

• The modulus of a function f(x) is

  • or

  • 

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: has solution 

    • But and 

    • So is not a solution to 

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