Question

graph the solution to the following inequality on the number line.

$x(x - 7) \leq 0$

Explanation:

Step1: Find the critical points

Set \(x(x - 7)=0\). Then \(x = 0\) or \(x-7=0\) (i.e., \(x = 7\)). These two points divide the number line into three intervals: \((-\infty,0)\), \((0,7)\), and \((7,\infty)\).

Step2: Test the intervals

• For the interval \((-\infty,0)\), let \(x=-1\). Then \(x(x - 7)=(-1)\times(-1 - 7)=(-1)\times(-8)=8>0\).
• For the interval \((0,7)\), let \(x = 1\). Then \(x(x - 7)=1\times(1 - 7)=1\times(-6)=-6<0\).
• For the interval \((7,\infty)\), let \(x = 8\). Then \(x(x - 7)=8\times(8 - 7)=8\times1 = 8>0\).

Step3: Consider the equality

Since the inequality is \(x(x - 7)\leq0\), the points \(x = 0\) and \(x = 7\) are included in the solution set (because when \(x = 0\) or \(x = 7\), \(x(x - 7)=0\)).

Answer:

The solution of the inequality \(x(x - 7)\leq0\) is the interval \([0,7]\). On the number - line, we draw a solid circle at \(x = 0\) and \(x = 7\) and shade the region between them.