Solve the following systems of linear in equations: 4/ x+1 ≤ 3 ≤ 6/x +1, x>0

Question

Solve the following systems of linear in equations:

\(\frac{4}{{\text{x}}+1}\) ≤ 3 ≤ \(\frac{6}{{\text{x}} +1}\), x > 0

Answer 1

 \(\frac{4}{{\text{x}}+1}\) ≤ 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