| x +1| less than or equal to | x - 3 | solve . please help

Question

solve it please

Answer 1

The absolute value in equality is |x + 1| ≤ |x - 3|.

There are four cases are obtained as follows :

Case 1 : (x + 1) > 0 and (x - 3) > 0

x + 1 ≤ x - 3 ⟹ 1 ≤ - 3 . This is false statement and there is no x - variable.

Case 2 : (x + 1) < 0 and (x - 3) < 0

- (x + 1) ≤ - (x - 3) ⟹ - x - 1 ≤ - x +3 ⟹ - 1 ≤ 3. This is true statement and there is no x - variable.

Case 3 : (x + 1) < 0 and (x - 3) > 0

- (x + 1) ≤ (x - 3) ⟹ - x - 1 ≤ x - 3 ⟹ 2 ≤ 2x ⟹ 1 ≤ x .

Case 4 : (x + 1) > 0 and (x - 3) < 0

(x + 1) ≤ - (x - 3) ⟹ x + 1 ≤ - x + 3 ⟹ 2x ≤ 2 ⟹ x ≤ 1.

The two solutions 1 ≤ x and x ≤ 1.