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The solution of the inequation \[\dfrac{{{\left( x-2 \right)}^{10000}}{{\left( x+1 \right)}^{253}}{{\left( x-\dfrac{1}{2} \right)}^{971}}{{\left( x+8 \right)}^{4}}}{{{x}^{500}}{{\left( x-3 \right)}^{75}}{{\left( x+2 \right)}^{93}}}\ge 0\] is?
\[\begin{align}
  & \left( A \right)\left( -\infty ,-2 \right)\cup{\left[ -1,0 \right)}\cup{\left( 0,\dfrac{1}{2} \right]}\cup{\left( 3,\infty \right)} \\
 & \left( B \right)\left( -\infty ,-2 \right]\cup{\left[ -1,0 \right)}\cup{\left[ 0,\dfrac{1}{2} \right]}\cup{\left[ 3,\infty \right)} \\
 & \left( C \right)\left( -\infty ,-2 \right]\cup{\left[ -1,0 \right]}\cup{\left( 0,\dfrac{1}{2} \right]}\cup{\left( 3,\infty \right)} \\
 & \left( D \right)\left( -\infty ,-2 \right)\cup{\left( -1,0 \right)}\cup{\left( 0,\dfrac{1}{2} \right)}\cup{\left( 3,\infty \right)} \\
\end{align}\]

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Last updated date: 14th Jun 2024
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Answer
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Hint: These types of problems are pretty straight forward and are very simple to solve. In problems such as these, we first analyse the problem and then try to find out all the possible critical points in the problem. These critical points mean all those points which may not lead to the satisfaction of the inequality. We consider such critical points and then try to eliminate them, so that the answers we get do not affect the value as well as the sign of the given inequality. Since in this problem the inequality given is greater than zero, so we consider only the terms whose power is odd.

Complete step by step solution:
Now, we start off with the solution of the given problem,
We first of all find all the critical points. Analysing closely, we find that the critical points in the given inequality are,
\[-2,-1,\dfrac{1}{2},3\]
Now, we know, that the right to \[3\] is positive infinity and the left to \[-2\] is negative infinity. Now if we draw a wavy curve or we take into consideration of the sign rule, we know that,
Right of \[3\] i.e. from \[3\] to positive infinity the sign of the inequality will be positive,
In between \[\dfrac{1}{2}\] and \[3\] the sign of the inequality will be negative,
In between \[-1\] and \[\dfrac{1}{2}\] the sign of the inequality will be positive,
In between \[-2\] and \[-1\] the sign of the inequality will be negative,
Left of \[-2\] i.e. from \[-2\] to negative infinity the sign of the inequality will be positive.
In the above findings, we only consider those which give a positive value, because the value of the inequality is positive. We also see that, in the positive range the value \[0\] also lies in it, which needs to be ignored, as it will lead to an undefined value.
Now writing the answer, we get
\[x\in \left( -\infty ,-2 \right)\cup{\left[ -1,0 \right)}\cup{\left( 0,\dfrac{1}{2} \right]}\cup{\left( 3,\infty \right)}\]

So, the correct answer is “Option A”.

Note: For these types of problems, we first need to keep in mind of all the rules and regulations to find the critical points. A fair knowledge of why to find critical points is also necessary. After finding the points, we need to check for the inclusive range, using the wavy curve method or the sign rule. Once done, we need to further check for points which may lead the inequality to be undefined.