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
Worked Example
Solve for :
a)

b) .

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
Examiner Tips and Tricks
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 .
a)

b)

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