2.9.3 Modulus Equations & Inequalities

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

DP IB Maths: AA HL

Revision Notes

2.9.3 Modulus Equations & Inequalities

Modulus Equations

How do I find the modulus of a function?

How do I solve modulus equations graphically?

How do I solve modulus equations analytically?

Worked Example

Modulus Inequalities

How do I solve modulus inequalities analytically?

Exam Tip

Worked Example

Author: Lucy

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