Question

solve for x and graph the solution. (x - 3)(x - 4) ≥ 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. submit

Explanation:

Step1: Find the critical points

Set \((x - 3)(x - 4)=0\). By the zero - product property, \(x-3 = 0\) gives \(x = 3\), and \(x - 4=0\) gives \(x = 4\). These are the critical points that divide the number line into three intervals: \((-\infty,3)\), \((3,4)\), and \((4,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,3)\), let \(x = 2\). Then \((2 - 3)(2 - 4)=(-1)\times(-2)=2>0\).
• For the interval \((3,4)\), let \(x=\frac{7}{2}\). Then \((\frac{7}{2}-3)(\frac{7}{2}-4)=(\frac{1}{2})\times(-\frac{1}{2})=-\frac{1}{4}<0\).
• For the interval \((4,\infty)\), let \(x = 5\). Then \((5 - 3)(5 - 4)=2\times1 = 2>0\).

Step3: Consider the equality

Since the inequality is \(\geq0\), the endpoints \(x = 3\) and \(x = 4\) are included in the solution set because when \(x = 3\) or \(x = 4\), \((x - 3)(x - 4)=0\).

Answer:

The solution of the inequality \((x - 3)(x - 4)\geq0\) is \(x\leq3\) or \(x\geq4\). On the number - line, we plot closed circles at \(x = 3\) and \(x = 4\). The graph consists of two rays: one from \(-\infty\) to \(3\) (including \(3\)) and another from \(4\) to \(\infty\) (including \(4\)).