Question

Solve the rational inequality and write the solution set in interval form.
$\dfrac{1}{x+10}>0$
a)$(-\infty ,10)$
b)$(10,-\infty )$
c)$(-10,\infty )$
d)\[\left[ 10,\infty \right]\]

Answer 1

Hint: In this question, we are given an inequality and we are asked to find out the interval in which x should lie so that the inequality is satisfied. Now, we know that a fraction will be greater than zero if its numerator and denominator are both positive or both negative. As here the numerator is one which is always positive, the inequality will be satisfied when the denominator will be positive and from there we can find the interval in which x should lie.

Complete step-by-step answer:
In this question, we are given a fraction and we should find the value of x for which it will be positive. Now, a fraction will be greater than zero if its numerator and denominator are both positive or both negative. As the numerator is 1, and 1 is always positive, , the inequality will be satisfied when the denominator will be positive. Therefore, the given inequality will be satisfied when
$x+10>0...............(1.1)$
Adding -10 to both sides of (1.1), we obtain
$\begin{align}
  & x>0-10 \\
 & \Rightarrow x>-10................(1.2) \\
\end{align}$
However, there is no restriction on the maximum value of x as the denominator will remain positive for any value of x greater than -10. Thus, we find that the inequality will be satisfied if the value of x is greater than -10 and less than infinity. In set theoretic form, we can write this as
$x\in \left( -10,\infty \right)$
Which matches option (c). Therefore, option (c) is the correct answer to this question.

Note: In the answer, we should note that that we should use open brackets $\left( -10,\infty \right)$ rather than closed brackets on both sides because at the right side, we should always use open brackets when using infinity and at the left side if x=-10, the denominator will become zero and hence the fraction will become undefined and thus the inequality will not be satisfied for x=-10.