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Algebra Level 5

$\large\dfrac{(\pi^x-7^x)\log_{10}(x-4)}{(x^2-9x+18)(x^2-x)} < 0$

If the solution of the above inequality is in the form $(a,b) \cup (c,\infty)$ , find $a+b+c$ .

1 solution

Akshay Yadav
May 17, 2016

$\dfrac{(\pi^x-7^x)\log_{10}(x-4)}{(x^2-9x+18)(x^2-x)} < 0$

Notice that $\log_{10}{(x-4)}$ is not defined $\forall x\leq 4$ .

Hence $x> 4$ ,

Simplifying the inequality,

$(\pi^x-7^x)\log_{10}(x-4)(x-6)(x-3)(x-1)x < 0$

Here, $x\neq 0,1,3,6$ .

Hence,

$x\in(4,5) \cup (6,+\infty)$

Nice solution, but another simpler method to check is wavy curve method

Ashish Menon - 4 years, 11 months ago

I agree with you (+1)

Akshat Sharda - 4 years, 11 months ago

$\log_{10}(x-4)$ is defined $x>4$ .

Akshat Sharda - 5 years ago

Yeah that's true.

Akshay Yadav - 5 years ago

You have a typo. You wrote $x \geq 4$ . It's supposed to only be $x > 4$

And in the table: $\pi^x - 7^x$ is negative for $(6,\infty)$

Hung Woei Neoh - 5 years ago

@Hung Woei Neoh – Thanks! I have made the relevant corrections.

Akshay Yadav - 5 years ago