Number of integers satisfying logx⁡$$4x+56$$−$$5x$$

Question

Number of integers satisfying logx⁡$$4x+56$$−$$5x$$<−$$1$$ is

Infinity Learn · Accepted Answer

logx⁡$$4x+56$$−$$5x$$<−$$1$$ We must have $$4x+56$$−$$5x$$>$$0$$⇒ $$4x+55x$$−$$6$$<$$0$$⇒ x∈−$$54$$,$$65$$ Also x>$$0$$ and x≠$$1$$∴ x∈$$0$$,$$65$$−$${1}-----(1)$$ Case I: $$01x$$⇒ $$4x+56$$−$$5x$$−$$1x$$>$$0$$⇒ $$4x2+5x+5x$$−$$6x(6$$−$$5x)$$>$$0$$⇒ $$4x2+10x$$−$$6x(5x$$−$$6)$$<$$0$$⇒ $$2(x+3)(2x$$−$$1)x(5x$$−$$6)$$<$$0$$⇒ x∈$$($$−$$3$$,$$0)$$∪$$12$$,$$65-----(3)$$ From (i), (ii) and (iii), we get ∴ x∈$$12$$,$$1$$ Case II: x>$$1----(4)logx$$⁡$$4x+56$$−$$5x$$<−$$1$$⇒ $$4x+56$$−$$5x$$<$$1x$$⇒ x∈$$($$−∞,−$$3)$$∪$$0$$,$$12$$∪$$65$$,∞$$-----(5)$$ From (i), (iv) and (v), we get x∈ϕ Thus, x∈$$12$$,$$1$$

$Number of integers satisfying \log_{x} \frac{4 x + 5}{6 - 5 x} < - 1 is$

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

Number of integers satisfying logx⁡4x+56−5x<−1 is

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

$Number of integers satisfying \log_{x} \frac{4 x + 5}{6 - 5 x} < - 1 is$