Inequalities Involving Absolute value

Inequalities Involving Absolute value

(i)  |x| ≤ a (where a > 0)

It implies those value of x on real number line which are at distance a or less than a from zero.

⇒ -a ≤ x ≤ a

Examples: |x| ≤ 2 ⇒ -2 ≤ x ≤ 2

|x| < 3 ⇒ -3 < x < 3

In general, |f(x)| ≤ a (where a > 0) ⇒ -a ≤ f(x) ≤ a

(ii)  |x| ≥ a (where a > 0)

It implies those value of x on real number line which are at distance a or more than a from zero.

⇒ x ≤ -a or x ≥ a

Examples: |x| ≥ 3 ⇒ x ≤ -3 or x ≥ 3

|x| > 2 ⇒ x < -2 or x > 2

In general, |f(x)| ≥ a ⇒ f(x) ≤ -a or f(x) ≥ a

(iii) a ≤ |x| ≤ b(where a, b > 0)

It implies those value of x on real number line whose distance from zero is equal to a or b or lies between a and b.

⇒ [-b, -a] ∪ [a, b]

Example: 2 ≤ |x| ≤ 4

⇒ x ϵ [-4, -2] [2, 4]

(iv) |x + y | < |x| + | y|, if x and y have opposite signs.

|x – y| < |x| – |y|, if x and y have same signs.

|x + y| = |x| + |y|, if x and y have same sign or at least one of x and y is zero.

|x – y| = |x| + |y|, if x and y have opposite sign or at least one of x and y is zero.