Question

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

Explanation:

Step1: Find critical points

Set each factor equal to zero: \(x - 1 = 0\) gives \(x = 1\); \(x - 2 = 0\) gives \(x = 2\). These are the critical points that divide the number line into intervals.

Step2: Test intervals

• For \(x < 1\) (e.g., \(x = 0\)): \((0 - 1)(0 - 2)=(-1)(-2)=2\geq0\), so this interval is part of the solution.
• For \(1 < x < 2\) (e.g., \(x = 1.5\)): \((1.5 - 1)(1.5 - 2)=(0.5)(-0.5)= - 0.25<0\), so this interval is not part of the solution.
• For \(x > 2\) (e.g., \(x = 3\)): \((3 - 1)(3 - 2)=(2)(1)=2\geq0\), so this interval is part of the solution.

Step3: Include critical points

Since the inequality is \(\geq0\), the critical points \(x = 1\) and \(x = 2\) are included in the solution.

Answer:

The solution to the inequality \((x - 1)(x - 2)\geq0\) is \(x\leq1\) or \(x\geq2\). To graph this, we plot closed circles at \(x = 1\) and \(x = 2\), draw a ray to the left of \(x = 1\) (including all values less than or equal to 1) and a ray to the right of \(x = 2\) (including all values greater than or equal to 2).