Question

solve for x and graph the solution. 2(x - 1)(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.

Explanation:

Step1: Find the roots

Set \(2(x - 1)(x - 4)=0\). Using the zero - product property \(a\times b\times c = 0\Rightarrow a = 0\) or \(b = 0\) or \(c = 0\). Here \(x-1=0\) gives \(x = 1\) and \(x - 4=0\) gives \(x = 4\). These roots divide the number line into three intervals: \((-\infty,1)\), \((1,4)\) and \((4,\infty)\).

Step2: Test intervals

• For \(x=0\) (in the interval \((-\infty,1)\)): \(2(0 - 1)(0 - 4)=2\times(-1)\times(-4)=8>0\).
• For \(x = 2\) (in the interval \((1,4)\)): \(2(2 - 1)(2 - 4)=2\times1\times(-2)=-4\leq0\).
• For \(x=5\) (in the interval \((4,\infty)\)): \(2(5 - 1)(5 - 4)=2\times4\times1 = 8>0\).

Since the inequality is \(2(x - 1)(x - 4)\leq0\), the solution includes the interval \([1,4]\) (the inequality is non - strict, so the endpoints \(x = 1\) and \(x = 4\) are included).

Answer:

The solution of the inequality \(2(x - 1)(x - 4)\leq0\) is \(x\in[1,4]\). On the number line, we plot a closed circle at \(x = 1\) and \(x = 4\) and shade the line segment between them.