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Solving modulus equations

Solving modulus equations

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Summary

Solving modulus equations

In a nutshell

You can solve modulus equations graphically and algebraically.



Solving a modulus equation

By understanding the domain, range, transformation and modulus of a function you can solve modulus equations. Most modulus functions have some sort of symmetry.


Example

Given the equation f(x)=2∣x−1∣−1f(x)=2\vert x-1\vert -1f(x)=2∣x−1∣−1, sketch a graph of y=f(x)y=f(x)y=f(x)​. Find its domain and range. Use this to find the points where the graph of y=f(x)y=f(x)y=f(x) intersects the graph of g(x)=x+1g(x)=x+1g(x)=x+1.


This function is like ∣x∣\vert x \vert∣x∣, but it has been vertically stretched with scale factor 222​, translated one unit to the right and then translated one unit downwards. So it looks like this:


Maths; Functions; KS5 Year 13; Solving modulus equations


Its domain is x∈R‾\underline{x\in \R}x∈R​ and its range is [−1,+∞)‾\underline{[-1,+\infty)}[−1,+∞)​.


Solving the equation can be done graphically or algebraically.


Maths; Functions; KS5 Year 13; Solving modulus equations


The two points of intersection are (0,1)(0,1)(0,1) and (4,5)(4,5)(4,5). To show this, create two equations: one where the argument of the modulus is positive and one where it is negative.

​​​​

​−2(x−1)−1=x+1−2x+2−1=x+1−2x+1=x+1x=0f(0)=1 ⟹ (0,1)‾\begin{aligned}-2(x-1)-1&=x+1\\-2x+2-1&=x+1\\-2x+1&=x+1\\x&=0\\f(0)=1&\implies \underline{(0,1)}\end{aligned}−2(x−1)−1−2x+2−1−2x+1xf(0)=1​=x+1=x+1=x+1=0⟹(0,1)​​​​
​2(x−1)−1=x+12x−2−1=x+12x−3=x+1x=4f(4)=5 ⟹ (4,5)‾\begin{aligned}2(x-1)-1&=x+1\\2x-2-1&=x+1\\2x-3&=x+1\\x&=4\\f(4)=5&\implies \underline{(4,5)}\end{aligned}2(x−1)−12x−2−12x−3xf(4)=5​=x+1=x+1=x+1=4⟹(4,5)​​​​


These are the points of intersection.





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FAQs - Frequently Asked Questions

How do I solve a modulus equation?

Form two equations: one where the arguement is positive and one where it is negative. Solve each equation individually.

How can I solve problems involving the modulus function?

First, sketch the function. Then form equations from the arguement. Solve the equations, algebraically, and check against the sketch.

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