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

Modulus Equations & Inequalities (DP IB Maths: AA HL)

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Lucy

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Modulus Equations

How do I find the modulus of a function?

The modulus of a function f(x) is

$|f (x)| = \{\begin{matrix} f (x) \\ - f (x) \end{matrix} \begin{matrix} f (x) \geq 0 \\ f (x) < 0 \end{matrix}$ or
$|f (x)| = \sqrt{[f ({x)]}^{2}}$

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: $x - 2 = 2 x - 3$ has solution $x = 1$
But $| (1) - 2 | = 1$ and $2 (1) - 3 = - 1$
So $x = 1$ is not a solution to $| x - 2 | = 2 x - 3$

Worked example

Solve for $x$ :

| \frac{2 x + 3}{2 - x} | = 5

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

|3 x - 1| = 5 x - 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 $x$ .

| 2 x - 1 | < 4

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

|x + 1| < |2 x + 3|

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

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Author: Lucy

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.