If the solution set of the inequality log x((5/2)-(1/x))1 is (a, b) ∪(c, d) then find the value of (c d/a b) where (a

Question

If the solution set of the inequality log x((5/2)-(1/x))>1 is (a, b) ∪(c, d) then find the value of (c d/a b) where (a < b < c < d). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Domain: x>(2/5) ; x ≠ 1 text Case-I: text If x>1 ⇒ (5/2)-(1/x)>x ⇒ x+(1/x)<(5/2) 2(x2+1)<5 x ⇒ 2 x2-5 x+2<0 2 x2-4 x-x+2<0

Q. If the solution set of the inequality $lo g_{x} (\frac{5}{2} - \frac{1}{x}) > 1$ is $(a, b) \cup (c, d)$ then find the value of $\frac{c d}{ab}$ where $(a < b < c < d) .$

Domain: $x > \frac{2}{5}; x \neq = 1$
$Case-I: If x > 1 \Rightarrow \frac{5}{2} - \frac{1}{x} > x \Rightarrow x + \frac{1}{x} < \frac{5}{2}$
$2 (x^{2} + 1) < 5 x \Rightarrow 2 x^{2} - 5 x + 2 < 0$
$2 x^{2} - 4 x - x + 2 < 0$

Q. If the solution set of the inequality logx​(25​−x1​)>1 is (a,b)∪(c,d) then find the value of abcd​ where (a<b<c<d).

Domain: x>52​;x=1 Case-I: If x>1⇒25​−x1​>x⇒x+x1​<25​ 2(x2+1)<5x⇒2x2−5x+2<0 2x2−4x−x+2<0

Q. If the solution set of the inequality $lo g_{x} (\frac{5}{2} - \frac{1}{x}) > 1$ is $(a, b) \cup (c, d)$ then find the value of $\frac{c d}{ab}$ where $(a < b < c < d) .$

Domain: $x > \frac{2}{5}; x \neq = 1$
$Case-I: If x > 1 \Rightarrow \frac{5}{2} - \frac{1}{x} > x \Rightarrow x + \frac{1}{x} < \frac{5}{2}$
$2 (x^{2} + 1) < 5 x \Rightarrow 2 x^{2} - 5 x + 2 < 0$
$2 x^{2} - 4 x - x + 2 < 0$