Question

The solution of the inequation $\frac{{(x-2)}^{10000}{(x+1)}^{253}{(x-\frac{1}{2})}^{971}{(x+8)}^4}{x^{500}{(x-3)}^{75}{(x+2)}^{93}}\ge 0$ is

• A. $(-\infty ,-2)\cup [-1,0)\cup (0,\frac{1}{2}]\cup (3,\infty )$
• B. $(-\infty ,-2]\cup [-1,0)\cup [0,\frac{1}{2}]\cup [3,\infty )$
• C. $(-\infty ,-2]\cup [-1,0]\cup (0,\frac{1}{2}]\cup [3,\infty )$
• D. $(-\infty ,-2)\cup (-1,0)\cup (0,\frac{1}{2})\cup (3,\infty )$

Answer 1

The correct option is A $(-\infty ,-2)\cup [-1,0)\cup (0,\frac{1}{2}]\cup (3,\infty )$

$\frac{{(x-2)}^{10000}{(x+1)}^{253}{(x-\frac{1}{2})}^{971}{(x+8)}^4}{x^{500}{(x-3)}^{75}{(x+2)}^{93}}\ge 0$

The critical points are $(-8),(-2),(-1),0,\frac{1}{2},2,3$ $[\because x\ne -2,0,3]$
The number line with given critical points:

Since, we required solution for greater than or equal to zero.
Hence, solution is $x\in (-\infty ,-2]\cup [-1,0)\cup (0,\frac{1}{2}]\cup (3,\infty )$