Question

Number of integral solutions of $$\left[ {\dfrac{{x + 2}}{{{x^2} + 1}}} \right] > \dfrac{1}{2}$$ is are
A.0
B.1
C.2
D.3
E.4

Answer 1

Hint: Here in this question, we need to find the integral solutions between the given inequality, for this, first we need to evaluate of simplify the given inequality by using a method of factorization further solve it for variable x then list out the integers lies between the bounds.

Complete step-by-step answer:
The solution such that all the unknown variables take only integer values is known as an integral solution.
Consider, the given inequality
$$\left[ {\dfrac{{x + 2}}{{{x^2} + 1}}} \right] > \dfrac{1}{2}$$ -----(1)
On cross multiplying, we have
$$ \Rightarrow 2\left( {x + 2} \right) > \left( {{x^2} + 1} \right)$$
$$ \Rightarrow 2x + 4 > {x^2} + 1$$
Subtract $$\left( {2x + 4} \right)$$ on both side, then we have
$$ \Rightarrow 0 > {x^2} + 1 - \left( {2x + 4} \right)$$
$$ \Rightarrow 0 > {x^2} + 1 - 2x - 4$$
$$ \Rightarrow 0 > {x^2} - 2x - 3$$
Or
$$ \Rightarrow {x^2} - 2x - 3 < 0$$
The above equation is similar to a quadratic equation $$a{x^2} + bx + c$$ now, solved by the method of factorization.
Now, Break the middle term as the summation of two numbers such that its product is equal to -3. Calculated above such two numbers are -3 and 1.
$$ \Rightarrow {x^2} - 3x + x - 3 < 0$$ ---(2)
Making pairs of terms in the above expression
$$ \Rightarrow \left( {{x^2} - 3x} \right) + \left( {x - 3} \right) < 0$$
Take out greatest common divisor GCD from the both pairs, then
$$ \Rightarrow x\left( {x - 3} \right) + 1\left( {x - 3} \right) < 0$$
Take $$\left( {x - 3} \right)$$ common
$$ \Rightarrow \left( {x - 3} \right)\left( {x + 1} \right) < 0$$
Equate the each factor to zero, then
$$ \Rightarrow \left( {x - 3} \right) = 0$$ or $$\left( {x + 1} \right) = 0$$
$$ \Rightarrow x = 3$$ $$x = - 1$$
The roots of $${x^2} - 3x + x - 3$$ is $$x = 3$$ and $$x = - 1$$.
So, the equation (2) becomes
$$ \Rightarrow - 1 < x < 3$$
Hence, the value of $$x$$ is 0, 1, 2.
$$\therefore $$ The number of integral solutions is 3.
Therefore, option (4) is the correct answer.
So, the correct answer is “Option 4”.

Note: An integral solution is a solution that all the unknown variables take only integer values. The equation is a quadratic equation. This problem can be solved by using the sum product rule. This defines as for the general quadratic equation $$a{x^2} + bx + c$$, the product of $$a{x^2}$$ and c is equal to the sum of bx of the equation. Hence, we obtain the factors. The factors for the equation depend on the degree of the equation.