Absolute (Modulus) Equations

Introduction

In mathematics, the absolute value of a number is the magnitude of that number, i.e. it's absolute distance from the origin.

Consider the following number line :

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Both $6$ and $-6$ have the exact same distance (absolute value) to the origin. The formal definition for the absolute value of a real number $x$ is :

$$|x|= \begin{cases} -x, & \text{if } x < 0, \\ x, & \text{if } x \ge 0. \end{cases}$$

In other words, we always take the positive value of the input of the modulus.

$$\begin{align*} |-3|=3 \\ |3|=3 \end{align*}$$

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Solving Modulus Equations

With the modulus of a real number defined, we now explore techniques into solving modulus equations.

Consider the simple case of :

$$|d|=5$$

Mathematically speaking, there exists some value $d$ such that the distance from the origin is 5 units. Intuitively, $d$ could either be $\pm5$.

$$d= \pm5$$

Another technique to get rid of the modulus is to square both sides because whether $d$ is positive or negative, the square of that value will always be positive.

$$\begin{align*} d^2&=25 \\ d&=\pm5 \end{align*}$$

Example

Now let's look at an alternative solution.

Example