Questions: Polynomial and Rational Functions Solving a quadratic inequality written in factored form Graph the solution to the following inequality on the number line. (x+5)(x-6)<0

Graph the solution to the inequality \((x+5)(x-6) < 0\) on the number line. Find the critical points

The critical points are where the expression \((x+5)(x-6)\) is equal to 0. Thus, we solve the equation \((x+5)(x-6) = 0\). This gives us \(x+5 = 0\) or \(x-6 = 0\). So, the critical points are \(x = -5\) and \(x = 6\). Determine the sign of the expression in each interval

We consider the intervals determined by the critical points: \(x<-5\), \(-5<x<6\), and \(x>6\).

• If \(x < -5\), then \(x+5 < 0\) and \(x-6 < 0\). Thus \((x+5)(x-6) > 0\).
• If \(-5 < x < 6\), then \(x+5 > 0\) and \(x-6 < 0\). Thus \((x+5)(x-6) < 0\).
• If \(x > 6\), then \(x+5 > 0\) and \(x-6 > 0\). Thus \((x+5)(x-6) > 0\).

Graph the solution on the number line

The solution to the inequality \((x+5)(x-6) < 0\) is the interval \(-5 < x < 6\). We represent this interval on the number line with an open circle at -5 and 6 and a line segment connecting them.

\(\boxed{(-5, 6)}\)

\((-5, 6)\)