3.7 Modulus function  (Page 2/2)

For the sake of understanding, we consider a non-negative number "2" equated to modulus of independent variable "x" like :

$$\left|x\right|=2$$

Then, the values of “x” satisfying this equation is :

$$\Rightarrow x=\pm 2$$

It is intuitive to note that values of "x" satisfying above equation is actually the intersection of modulus function "y=[x]" and "y=2" plots as shown in the figure.

Further, it is easy to realize that equating a modulus function to a negative number is meaningless. The modulus expression "|x|" always evaluates to a non-negative number for all real values of "x". Observe in the figure above that line "y=-2" does not intersect modulus plot at all.

We express these results in general form, using an expression f(x) in place of "x" as :

$$\left|\mathrm{f(x)}\right|=a;a>0\Rightarrow \mathrm{f(x)}=\pm a$$

$$\left|\mathrm{f(x)}\right|=a;a=0\Rightarrow \mathrm{f(x)}=0$$

$$\left|\mathrm{f(x)}\right|=a;a<0\Rightarrow \text{There is no solution of this equality}$$

Modulus as distance

The modulus of an expression “x-a” is interpreted to represent “distance” between “x” and “a” on the real number line. For example :

$$|x-2|=5$$

This means that the variable “x” is at a distance “5” from “2”. We see here that the values of “x” satisfying this equation is :

$$\Rightarrow x-2=\pm 5$$

Either

$$\Rightarrow x=2+5=7$$

or,

$$\Rightarrow x=2-5=-3$$

The “x = 7” is indeed at a distance “5” from “2” and “x=-3” is indeed at a distance “5” from “2”. Similarly, modulus |x|= 3 represents distance on either side of origin.

Modulus and inequality

Interpretation of inequality involving modulus depends on the nature of number being compared with modulus.

Case 1 : a>0

The values of x that satisfy "less than (<)" inequality lies between intervals defined between -a and a, excluding end points of the interval. On the other hand, values of x that satisfy "greater than (>)" inequality lies in two disjointed intervals.

$$\left|x\right|<a;a>0\Rightarrow -a<x<a$$

$$\left|x\right|>a;a>0\Rightarrow x<-a\mathrm{or}x>a\Rightarrow x\in \left(\mathrm{-\infty }\mathrm{,a}\right)\cup \left(\mathrm{a,}\infty \right)$$

Important aspect of these inequalities is that they can be used to express intervals in compact form. For example, range of cosecant trigonometric function is $x\in \left(\mathrm{-\infty }\mathrm{,-1}\right]\cup \left[\mathrm{1,}\infty \right\}$ . Equivalently, we can write this interval as |x|≥1.

We extend these results to an expression as :

$$\left|\mathrm{f(x)}\right|<a;a>0\Rightarrow -a<\mathrm{f(x)}<a$$

For inequality involving greater than comparison with a positive number represents union of two separate intervals.

$$\left|\mathrm{f(x)}\right|>a;a>0\Rightarrow \mathrm{f(x)}<-a\text{or}\mathrm{f(x)}>a$$

Case 2 : a<0

Modulus can not be equated to negative number as modulus always evaluates to non-negative number. Clearly, modulus of an expression or variable can not be less than a negative number. However, modulus function is always greater than negative number. Hence, we conclude that :

$$\left|x\right|<a;a<0\Rightarrow \text{There is no solution of this inequality}$$

$$\left|x\right|>a;a<0\Rightarrow \text{This inequality is valid for all real values of x}$$

Extending these results to expression, we have :

$$\left|\mathrm{f(x)}\right|<a;a<0\Rightarrow \text{There is no solution of this inequality}$$

$$\left|\mathrm{f(x)}\right|>a;a<0\Rightarrow \text{This inequality is valid for all real values of f(x)}$$

Additional properties of modulus function

Here, we enumerate some more properties of modulus of a real variable i.e. modulus of a real number.

Let x,y and z be real variables. Then :

$$|-x|=\left|x\right|$$ $$|x-y|=0\iff x=y$$ $$|x+y|\le \left|x\right|+\left|y\right|$$ $$|x-y|\ge \left|\right|x|-|y\left|\right|$$ $$\left|xy\right|=\left|x\right|X\left|y\right|$$ $$\left|\frac{x}{y}\right|=\frac{\left|x\right|}{\left|y\right|};\left|y\right|\ne 0$$

Modulus |x-y| represents distance of x from y. Also, we know that sum of two sides of a triangle is greater than third side. Combining these two facts, we write a general property for modulus involving real numbers as :

$$|x-y|<|x-z|+|z-y|$$

Square function

There is striking similarity between modulus and square function. Both functions evaluate to non-negative values.

$$y=\left|x\right|;y\ge 0$$

$$y=x^2;y\ge 0$$

Their plots are similar. Besides, they behave almost alike to equalities and inequalities. We shall not discuss each of the cases as done for the modulus function, but with a specific number (4 or -4). We shall enumerate each of the possibilities, which can be easily understood in the background of discussion for modulus function.

1: Equality

$$x^2=4\Rightarrow x=\pm 2$$

$$x^2=-4\Rightarrow \text{No solution}$$

2: Inequality with non-negative number

A. Less than or less than equal to

$$x^2<4\Rightarrow -2<x<2$$

B. Greater than or greater than equal to

$$x^2>4\Rightarrow x<-2\text{or}x>2\Rightarrow (-\infty ,-2)\cup \left(\mathrm{2,}\infty \right)$$

3: Inequality with negative number

A. Less than or less than equal to

$$x^2<-4\Rightarrow \text{No solution}$$

B. Greater than or greater than equal to

$$x^2>-4\Rightarrow \text{Always true}$$

Acknowledgment

Author wishes to thank Mr. Ritesh Shah for making suggestion to remove error in the module.