Question

((x-5)*(x+3))/(x-1) less than/equal to 0 Solve the inequality

Answer 1

The way to solve this problem is to find the boundaries and then evaluate the expression in the intervals created by the boundaries to see if the inequality is satisfied or not in each interval. Start by examining the equality.
\[(x-5)(x+3)/(x-1) = 0\]
Now use the zero product property to say that \[(x-5) = 0 \] or \[(x+3) = 0\]
Solving those two gives you solutions to the equalities of x = 5 and x = -3. Put those points on a number line.
Now fine the disallowed value by setting the denominator equal to zero. That tells you that
\[x \neq1\]
So, put an open circle at 1 on your number line. Now, you have two points at 5 and -3, and one open circle at 1. Those three divide your number line into four intervals. Now, you just test each interval by inspection to see if the expression on the left side of your original inequality is greater than zero or not. It's easiest to start at the boundary and move away from it. For example, at -3, the expression is exactly zero because you would have -3 + 3 in the (x-3) term. As x gets more negative, that factor is negative. Do that for the other two factors and you find you get (-)(-)/(-) which is negative, so that interval is not part of the solution. Do that procedure for the other three intervals and you'll find the solution is \[-3\le x \le 1\] and\[x \ge 5\]