Question

solve for x and graph the solution.
4(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.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Explanation:

Step1: Find the critical points

Set \(4(x - 3)(x - 5)=0\). Using the zero - product property \(a\times b\times c = 0\) implies \(a = 0\) or \(b = 0\) or \(c = 0\). Here, \(x-3=0\) 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 \((5,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,3)\), let \(x = 0\). Then \(4(0 - 3)(0 - 5)=4\times(-3)\times(-5)=60>0\).
• For the interval \((3,5)\), let \(x = 4\). Then \(4(4 - 3)(4 - 5)=4\times1\times(-1)=-4\leq0\).
• For the interval \((5,\infty)\), let \(x = 6\). Then \(4(6 - 3)(6 - 5)=4\times3\times1 = 12>0\).

Step3: Consider the equality case

Since the inequality is \(\leq0\), the values of \(x\) that make \(4(x - 3)(x - 5)=0\) (i.e., \(x = 3\) and \(x = 5\)) are included in the solution set.

Answer:

The solution of the inequality \(4(x - 3)(x - 5)\leq0\) is \(3\leq x\leq5\). On the number - line, we plot closed circles at \(x = 3\) and \(x = 5\) and shade the line segment between them.