Logarithmic Inequality

Algebra Level 5

Find all values of the parameter $a \in \mathbb{R}$ for which the following inequality is valid for all $x \in \mathbb{R}:$

$1+\log _{ 5 }{ { (x }^{ 2 }+1) } \ge \log _{ 5 }{ (a{ x }^{ 2 }+4x+a) }.$

If the range of values of $a$ can be expressed in the form of $(A,B],$ then find the value of $A + B.$

1 solution

Tanishq Varshney
Apr 2, 2015

$(a-5)x^2+4x+(a-5) \leq 0$ $\quad$ and $\quad$ $ax^2+4x+a >0$

$a-5<0$ Its discriminant $D \leq 0$ $\quad \quad \quad \quad \quad$ its $D <0$

$16-4(a-5)^{2} \leq 0$ $\quad \quad \quad \quad \quad$ $a^2-4 >0$

$(a-7)(a-3) \geq 0$ $\quad \quad \quad \quad \quad$ $(a-2)(a+2)>0$

The solution is $(2,3]$

The above solution is incomplete. You didn't mention the conditions, $a-5<0$ for first inequality and $a>0$ for the second one.

Abhishek Sharma - 6 years, 2 months ago

Sorry i forgot to do that. edited thanx

Tanishq Varshney - 6 years, 2 months ago

The solution is perfect now. +1

Abhishek Sharma - 6 years, 2 months ago

Do u have any suggestion about this

Tanishq Varshney - 6 years, 2 months ago

Can you tell me how you know a-5 < 0 and D <= 0?

JJ Chai - 5 years, 2 months ago

Just basic quadratic equations. Refer a book.

Abhishek Sharma - 5 years, 2 months ago