Question

solve the following inequality and graph the solution:

$(x - 2)(x - 4) > 0$

choose test values and indicate whether the inequality is true or false in each region.

write the solution as a compound inequality

and draw the solution

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Explanation:

Step1: Find the critical points

Set \((x - 2)(x - 4)=0\). Using the zero - product property \(x-2 = 0\) gives \(x = 2\), and \(x - 4=0\) gives \(x = 4\). These points divide the number line into three intervals: \((-\infty,2)\), \((2,4)\), and \((4,\infty)\).

Step2: Test values in each interval

• For the interval \((-\infty,2)\), let \(x=1\). Then \((1 - 2)(1 - 4)=(-1)\times(-3)=3>0\).
• For the interval \((2,4)\), let \(x = 3\). Then \((3 - 2)(3 - 4)=(1)\times(-1)=-1<0\).
• For the interval \((4,\infty)\), let \(x=5\). Then \((5 - 2)(5 - 4)=(3)\times(1)=3>0\).

Step3: Write the solution as a compound inequality

The inequality \((x - 2)(x - 4)>0\) is true for \(x<2\) or \(x>4\). In compound - inequality form, it is \(x<2\) or \(x>4\).

Answer:

The solution of the inequality \((x - 2)(x - 4)>0\) is \(x<2\) or \(x>4\). On the number line, we draw an open circle at \(x = 2\) and \(x = 4\) (since the inequality is strict, i.e., \(>\) not \(\geq\)) and shade the regions to the left of \(x = 2\) and to the right of \(x = 4\).