\(\frac{4}{{\text{x}}+1}\) ≤ 3 ≤ \(\frac{6}{{\text{x}} +1}\), x > 0
When,
\(\frac{4}{{\text{x}}+1}\) ≤ 3
Subtracting 3 from both the sides
Signs of 1 – 3x:
1 – 3x = 0 → \({\text{x}} = \frac{1}{3}\)
(Subtract 1 from both the sides and then divide both sides by -3)
1 – 3x = 0 → \({\text{x}} < \frac{1}{3}\)
(Subtract 1 from both the sides, then multiply by -1 on both sides and then divide both sides by 3)
1 – 3x = 0 → \({\text{x}} > \frac{1}{3}\)
(Subtract 1 from both the sides, then multiply by -1 on both sides and then divide both sides by 3)
Interval satisfying the required condition ≤ 0 , x > 0
Subtracting 3 from both the sides
Dividing both sides by 3
Multiplying by -1 on both sides
Signs of x - 1:
x – 1 = 0 → x = 1 (Adding 1 to both the sides)
x – 1 > 0 → x > 1 (Adding 1 to both the sides)
x – 1 < 0 → x < 1 (Adding 1 to both the sides)
Interval satisfying the required condition: ≤ 0
x ≤ 1
Combining the intervals:
\(\frac{1}{3} \le {\text{x}} <1\) such that x>0