Question

Solve each inequality. Graph the solution set.

$\frac{(x-2)(x-1)}{x-3}\ge 0$

Short answer

The real zeroes of the function are 1,2 and 3 and the solution set can be graphed as

Step-by-step solution

Step 1. Given Information 

The given inequality is$\frac{(x-2)(x-1)}{(x-3)}\ge 0$

Step 2. Calculation

The inequality is already in a form with zero on the right side.

Solve the given inequality by finding the real zeroes of the numerator and denominator.

$$\begin{gathered} x-2=0\Rightarrow x=2 \\ x-1=0\Rightarrow x=1 \\ x-3=0\Rightarrow x=3 \end{gathered}$$

Thus, real zeroes of the function are 1, 2 and 3.

Step 3. Graph 

Construct the table by using a number from each interval to check whether the function is positive or negative.

Thus, the solution set is $\left\{x|1\le x\le 2orx>3\right\}$

And the solution set can be graphed as