Question

Solve: { [x+3] +x/x+2} > 1.​

Answer 1

Step-by-step explanation:

We have (|x+3|+x)/(x+2)>1 Now two cases arise: Case I When x + 3 ≥ 0, i.e., x ≥ – 3. Then ⇒ {(x + 1) > 0 and x + 2 > 0} or {x + 1 < 0 and x + 2 < 0} ⇒ {x > –1 and x > –2} or {x < – 1 and x < – 2} ⇒ x > –1 or x < – 2 ⇒ x ∈ (–1, ∞) or x ∈ (– ∞, – 2) ⇒ x ∈ (–3, –2) ∪ ( – 1, ∞) [Since x ≥ – 3] ... (1) Case II When x + 3 < 0, i.e., x < – 3 ⇒ (x + 5 < 0 and x + 2 > 0) or (x + 5 > 0 and x + 2 < 0) ⇒ (x < – 5 and x > – 2) or (x > – 5 and x < –2) it is not possible. ⇒ x ∈ (– 5 , – 2) ... (2) Combining (I) and (II), the required solution is x ∈ (– 5 , – 2) ∪ (– 1, ∞)