Solve the given nonlinear inequality. Write the solution set using interval notation. Graph the solution set.$x^2-8x+12<0$

Key Concepts: Quadratic Inequalities, Interval Notation, Graphing Explanation: To solve the inequality, we first need to find the critical points by setting the left side equal to zero and factoring. This gives us:\n $x^2-8x+12=0$\n $(x-2)(x-6)=0$\n So the critical points are $x=2$ and $x=6$. We can now use these critical points to divide the number line into three intervals: $(-\infty ,2)$, $(2,6)$, and $(6,\infty )$.\n We now need to test each interval to see if it satisfies the inequality. For the interval $(-\infty ,2)$, we can choose any value less than 2 and plug it into the inequality. Let's choose $x=0$:\n $0^2-8(0)+12<0$\n $12>0$, which is false. So we know that $(-\infty ,2)$ does not satisfy the inequality.\n For the interval $(2,6)$, we can choose any value between 2 and 6. Let's choose $x=4$:\n $4^2-8(4)+12<0$\n $-4<0$, which is true. So we know that $(2,6)$ satisfies the inequality.\n For the interval $(6,\infty )$, we can choose any value greater than 6. Let's choose $x=8$:\n $8^2-8(8)+12<0$\n $-4<0$, which is true. So we know that $(6,\infty )$ satisfies the inequality.\n Therefore, the solution set is $(2,6)\text{ or }2<x<6$, and we can graph this on the number line. Step by Step Solution: Step 1. Find the critical points by setting the left side equal to zero and factoring: $x^2-8x+12=0$ $(x-2)(x-6)=0$So the critical points are $x=2$ and $x=6$. Step 2. Divide the number line into three intervals: $(-\infty ,2)$, $(2,6)$, and $(6,\infty )$. Step 3. Test each interval to see if it satisfies the inequality:\n- For the interval $(-\infty ,2)$, choose any value less than 2 and plug it into the inequality.\n- For the interval $(2,6)$, choose any value between 2 and 6 and plug it into the inequality.\n- For the interval $(6,\infty )$, choose any value greater than 6 and plug it into the inequality. Step 4. The solution set is $(2,6)\text{ or }2<x<6$. Final Answer: $(2,6)\text{ or }2<x<6$