Question

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

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x = 0\) and \(x - 1 = 0\) (so \(x = 1\)). These are the critical points that divide the number line into intervals.

Step2: Test intervals

• For \(x < 0\) (e.g., \(x = -1\)): \((-1)(-1 - 1)=(-1)(-2)=2>0\), so the inequality is not satisfied here.
• For \(0 < x < 1\) (e.g., \(x = 0.5\)): \((0.5)(0.5 - 1)=(0.5)(-0.5)= - 0.25\leq0\), so the inequality is satisfied here.
• For \(x > 1\) (e.g., \(x = 2\)): \((2)(2 - 1)=(2)(1)=2>0\), so the inequality is not satisfied here.

Step3: Include critical points

Since the inequality is \(\leq0\), we include the critical points \(x = 0\) and \(x = 1\) where the expression equals zero.

Answer:

The solution to the inequality \(x(x - 1)\leq0\) is \(0\leq x\leq1\). To graph this, plot closed circles at \(x = 0\) and \(x = 1\) and draw a line segment connecting them on the number line.