Question

solve for x and graph the solution.
-(x + 3)(x - 5) ≤ 0
plot the endpoints. select an endpoint to change it from closed to open. select the middle of a segment, ray, or line to delete it.

Explanation:

Step1: Multiply both sides by -1

When multiplying an inequality by a negative number, the direction of the inequality sign changes. So, \(-(x + 3)(x - 5)\leq0\) becomes \((x + 3)(x - 5)\geq0\).

Step2: Find the critical points

Set each factor equal to zero: \(x+3 = 0\) gives \(x=-3\), and \(x - 5=0\) gives \(x = 5\). These are the critical points that divide the number line into three intervals: \((-\infty,-3)\), \((-3,5)\), and \((5,\infty)\).

Step3: Test intervals

• For \(x<-3\) (e.g., \(x=-4\)): \((-4 + 3)(-4 - 5)=(-1)\times(-9)=9\geq0\).
• For \(-3
• For \(x>5\) (e.g., \(x=6\)): \((6 + 3)(6 - 5)=(9)\times(1)=9\geq0\).

Step4: Consider the equality

Since the original inequality is \(\geq\), the endpoints \(x=-3\) and \(x = 5\) are included in the solution.

Answer:

The solution is \(x\leq - 3\) or \(x\geq5\). On the number - line, plot a closed circle at \(x=-3\) and shade to the left, and plot a closed circle at \(x = 5\) and shade to the right.