Question

Solve the following inequation and represent your solution on the real number line: −5 1 2 − x ≤ 1 2 − 3x ≤ 3 1 2 − x, x ∈ R

Answer 1

CASE I :- When x < 0
In this case |x| = -x --------------> Therefore - x > a -----------> x < -a
Thus in this case the solution set of the given Inequation is given by
x < 0 and x < -a ---------------> x < -a

Combining two cases, we get x > a or x < -a


Result 3 :- For |x| < 0 -----> No Solution.


Result 4 :- For |x| ≥ 0 ------> -∞ < x < ∞ (all real numbers)


Here are another two results which are derived from above basic results

Result 5 :- Let r be positive real and a be a fixed number, then |x - a|< r ------> (a – r) < x < (a + r)
We know that (From Result 1) |x| < a = -a < x < a
We have |x - a|< r ------------> -r < x-a < r ----------> a – r < x < a + r


Result 6 :- Let r be positive real and a be a fixed number, then |x - a|> r ------> x < a – r or x > a + r
We know that (From Result 2) |x| > a = x > a or x < -a
We have |x - a|> r ------------> x – a > r -------> x > a + r ----------or ---------> x –a < - r -------> x < a – r


Example 1 :- |3x-2| ≤ 1212

We know that |x-a| ≤ r = a - r ≤ x ≤ a + r

|3x-2| ≤ 1212 ------> 2 - 1212 ≤ 3x ≤ 2 + 1212 ------> 3232 ≤ 3x ≤ 5252 --------> 1212 ≤ x ≤ 5656 -----> [ 1212, 5656 ]




Example 2 :- |x - 2| ≥ 5
|x - a| ≥ r = x ≥ a + r or x ≤ a – r
|x – 2| ≥ 5 -------> x ≥ 2 + 5 or x ≤ 2 – 5 -----------> x ≥ 7 or x ≤ -3 --------> (-∞, -3] or [7, ∞)