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# Absolute Value

As we have observed that there is an order preserving one-to-one correspondence between elements of R and points on the number line.

Absolute Value

## 1. 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 0 on the number line is called the absolute value of that number a and is denoted by |a|. Thus, for any x R, we have and hence |·| defines a function known as absolute value function, from R onto [0, ∞). ## 2. 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. ## 3. Some Results For Absolute Value

(i) If x, y R, |y + x| = |x − y|, then xy = 0.

(ii) For any x, y R, |xy| = |x||y|.

(iii) |x/y| = |x/y|, for all x, y R and y ¹ 0.

(iv) For any x, y R, |x + y|≤|x| + |y|.

## 4. 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 > 0 as |x| ≥ 0.

There are two possibilities to consider depending on the sign of x.

Case (1). If x ≥ 0, then |x| = x, so |x| < r implies x<r.

Case (2). If x < 0, 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).

(ii) Let us prove that |x| > r if and only if x < −r or x>r.

Consider |x| > r. If r < 0, then every x R satisfies the inequality.

For r ≥ 0, there are two possibilities to consider.

Case (1). If x ≥ 0, then |x| = x>r.

Case (2). If x < 0, then |x| = −x>r, that is, x < −r.

So we have |x| > r, if and only if x < −r or x > r, that is, x (−∞, −r) (r, ∞).

Remark:

(i) For any a R, |x − a| ≤ r if and only if −r ≤ x − a ≤ r if and only if x [a − r, a + r].

(ii) For any a R, |x − a| ≥ r is equivalent to x − a ≤ −r or x − a ≥ r if and only if x (−∞, a − r] [a + r, ∞).  Tags : Definition, Properties, Equations Formula, Solved Example Problems, Exercise | Algebra | Mathematics , 11th Mathematics : UNIT 2 : Basic Algebra
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11th Mathematics : UNIT 2 : Basic Algebra : Absolute Value | Definition, Properties, Equations Formula, Solved Example Problems, Exercise | Algebra | Mathematics