Problem 32 - Algebra and Trigonometry

Question

Solve each polynomial inequality in Exercises \(1-42\) and graph the solution set on a real number line. Express each solution set in interval notation. $$ x(4-x)(x-6) \leq 0 $$

Step-by-Step Solution

Answer

The solution of the polynomial inequality \(x(4-x)(x-6) \leq 0\) is \(x \in [0,4] \cup [6,\infty)\

Step 1: Solve the equation

Firstly, set the inequality to be an equality i.e \(x(4-x)(x-6) = 0 \). To find the roots of the polynomial, set each factor to zero and solve for \(x\). That gives \(x = 0,4,6\). Now, \(x = 0\), \(x = 4\), and \(x = 6\) are critical points that divide the number line into four intervals

Step 2: Define the intervals

The four intervals are: \[(-\infty,0)], \((0,4)\), \((4,6)\), and \((6,\infty)\). Now we need to check these intervals in the inequality and determine which ones satisfy the condition.

Step 3: Test the sign of the polynomial on each interval

A good choice for test point in the interval \[(-\infty,0)\] is -1. When we substitute -1 into the inequality \(x(4-x)(x-6) \leq 0\) , we get a positive number. Thus, the interval \[(-\infty,0)\] isn't part of the solution. Next, substitute 2 as a representative of the interval \((0,4)\). The inequality evaluates to a negative number, so \((0,4)\) is part of the solution. Similarly, choose 5 as a representative of interval \((4,6)\). Substituting gives a positive number, so \((4,6)\) isn't part of the solution. Lastly, substitute 7 for \((6,\infty)\). The inequality evaluates to a negative number, so \((6,\infty)\) is part of the solution.

Step 4: Express solution set in interval notation

Interval notation is just another way of writing the solution to an inequality. The solution to the inequality \(x(4-x)(x-6) \leq 0\) is the union of the intervals \((0,4)\) and \((6,\infty)\), together with the points 0,4,6 where the polynomial is 0, because the inequality includes equals. Therefore, the solution is \(x \in [0,4] \cup [6,\infty)\