Question

solve for x and graph the solution.
5x(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 critical points

Set \(5x(x - 4)=0\). Using the zero - product property \(a\times b = 0\) implies \(a = 0\) or \(b = 0\).
For \(5x=0\), we get \(x = 0\). For \(x - 4=0\), we get \(x = 4\).

Step2: Test intervals

We have three intervals to test: \((-\infty,0)\), \((0,4)\), and \((4,\infty)\).

• For the interval \((-\infty,0)\), let \(x=-1\). Then \(5\times(-1)\times(-1 - 4)=5\times(-1)\times(-5)=25>0\).
• For the interval \((0,4)\), let \(x = 1\). Then \(5\times1\times(1 - 4)=5\times1\times(-3)=-15<0\).
• For the interval \((4,\infty)\), let \(x = 5\). Then \(5\times5\times(5 - 4)=25\times1 = 25>0\).

Since the inequality is \(5x(x - 4)\leq0\), the solution includes the values of \(x\) for which the expression is non - positive. Also, since the inequality is \(\leq\) (not just \(<\)), the critical points \(x = 0\) and \(x = 4\) are included in the solution.

Answer:

The solution of the inequality \(5x(x - 4)\leq0\) is \(0\leq x\leq4\). On the number line, we plot closed circles at \(x = 0\) and \(x = 4\) and shade the segment between them.