Question

Find the values of x for (x+3)(x-1)/(x+2)(x-2)>and equal to 0

Answer 1

\[\frac{ \left( x+3 \right)\left( x-1 \right) }{ \left( x+2 \right)\left( x-2 \right) }\ge 0\]
\[ clearly ,x \neq -2,x \neq 2\]
\[\frac{ x ^{2}+3x-x-3 }{x ^{2}+2x-2x-4 }=\frac{ x ^{2}+2x-3 }{ x ^{2}-4 }\]
\[\frac{ x ^{2}+2x+1-1-3 }{ x ^{2} -4}=\frac{ \left( x+1 \right)^{2}-2^{2} }{ x ^{2}-2^{2} }\]
either numerator and denominator both are positive or both are negative
case1, both are positive
\[\left( x+1 \right)^{2}-2^{2}\ge0,\left| x+1 \right|\ge2\]
\[x+1\le-2,x \le -2-1,x \le -3\]
\[and x+1\ge 2,x \ge 2-1,x \ge 1 but x \neq 2 x \in [1,2)\cup (2,\infty) \]
\[\left| x \right|\le-2,and x \ge2\]
\[combining x \in (-\infty,-3]\cup(2,\infty )...(1)\]
case2
\[\left( x+1 \right)^{2}-2^{2}<0,\left| x+1 \right|<2,-2<x+1 <2,-3<x <1,but x \neq-2\]
\[x \in (-3,-2)\cup (-2,1)\]
\[x ^{2}-2^{2}<0,\left| x \right|<2,-2<x <2\]
\[combining, x \in (-2,1)... (2)\]
(1) and (2) are the solutions.
if you find any difficulty in understanding please feel free to ask.