Question

How do you solve and write the following in interval notation:- $x - \dfrac{{48}}{x} < \- 8$?

Answer 1

Simplify and solve the quadratic equation to get the answer
The very first step to solving this problem will be to simplify the problem and write the given question in a quadratic format. After writing the above problem in quadratic format, you will need to solve the above problem for x by removing the inequality symbol by equal sign, which will give you the critical points. Use the original inequality symbol then to find the final answer.

Complete step by step solution:
The given question we have is $x - \dfrac{{48}}{x} < \- 8$
Simplifying the equation, we will get:-
$\dfrac{{{x^2} - 48}}{x} < \- 8 \\\ \Rightarrow {x^2} - 48 < \- 8x \\\$
Now, adding$8x$on both sides of the equation, we will get:-
${x^2} - 48 + 8x < 0$
Now, replace the lesser than symbol with equal to. We will use splitting the middle term method to solve the above problem.
${x^2} + 12x - 4x - 48 = 0 \\\ \Rightarrow x(x + 12) - 4(x + 12) = 0 \\\ \Rightarrow (x - 4)(x + 12) = 0 \\\$
Now, since$(x - 4)(x + 12) = 0$. Either $(x - 4) = 0$or$(x + 12) = 0$
Therefore, $( - 12,4)$and $x = - 12$ is the answer for the quadratic equation
Now, we will again use the original symbol. That is lesser than instead of the equal sign. Which will give us:-
$x < 4$and $x < \- 12$
Hence, if we try to explain this answer in words, we will get
X can be any real number on the number line in between 4 and -12. The end points are not included.
Therefore, we will write the given answer in the interval format as
Interval= $( - 12,4)$

Note: We ignore the equal to sign only to solve the equation easily. Always remember, whenever you have to solve an inequality where a quadratic equation is there. You will need to ignore the symbol for a few steps and start solving it normally. After getting the critical points, use the original symbol to find the required intervals.