Absolute Value

. . Definition and Properties As we have observed that there is an order preserving one-to-one correspondence between elements of R and points on the number line. Note that for each x ∈ R , x and − x are equal distance from the origin.

The distance of the number a ∈ R from on the number line is called the absolute value of the number a and is denoted by | a | . Thus, for any x ∈ R , we have | x | = if x ≥ , − x if x < . and hence | · | defines a function known as absolute value function, from R onto [ , ∞ ) and the graph of this function is discussed in Chapter . Basic Algebra (i) For any x ∈ R , we have | x | = | − x | and thus, | x | = | y | if and only if x = y or x = − y .

(ii) | x − a | = r if and only if r ≥ and x − a = r or x − a = − r . . . Equations Involving Absolute Value Note that a real number a is said to be a solution of an equation or an inequality, if the statement obtained after replacing the variable by a is true.

Next we shall learn solving equations involving absolute value. Example . Solve | x − | = for x. | x − | = .

Then, we have x − = ± which implies x = or x = . Example . Solve | x − | + = for x. | x − | + = .

So that we have, | x − | = − = . Thus, we have either x − = or x − = − . Therefore the solutions are x = − and x = . Example .

Solve | x − | = | x − | . We know that | u | = | v | if and only if u = v or u = − v . Therefore, | x − | = | x − | implies x − = x − or x − = − x . Solving these two equations, we get x = − and x = .

Hence, both x = − and x = are solutions. . . Some Results For Absolute Value (i) If x, y ∈ R , | y + x | = | x − y | , then xy = .

(ii) For any x, y ∈ R , | xy | = | x || y | . (iii) = | x | | y | , for all x, y ∈ R and y = . (iv) For any x, y ∈ R , | x + y | ≤| x | + | y | . .

. Inequalities Involving Absolute Value Here we shall learn to solve inequalities involving absolute values. First we analyze very simple inequalities such as (i) | x | < r and (ii) | x | > r. (i) Let us prove that | x | < r if and only if − r < x < r .

Note that r > as | x | ≥ . There are two possibilities to consider depending on the sign of x. Case ( ). If x ≥ , then | x | = x, so | x | < r implies x < r .

Case ( ). If x < , then | x | = − x , so | x | < r implies − x < r that is, x > − r. Therefore we have, | x | < r if and only if − r < x < r, that is x ∈ ( − r, r ) . .