Problem 31 - 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. $$ x(3-x)(x-5) \leq 0 $$

Step-by-Step Solution

Answer

The solution set in interval notation is \([0, 3]\).

Step 1: Find Critical Points

Solve the equation \(x(3-x)(x-5) = 0\) for x. This yields the points 0, 3, and 5. These are the critical points.

Step 2: Setup the Number Line

Place these critical points on a number line, which creates four intervals. Those intervals are \(-\infty, 0\), \(0, 3\), \(3, 5\), and \(5, \infty\).

Step 3: Testing Each Interval

Since the inequality includes equals to 0, include the critical points in the solution. Test a number in each interval to see if it satisfies the inequality. For \(-\infty, 0\), let's pick \(-1\), the inequality simplifies to a negative, thus this interval is not a part of the solution. For \(0, 3\), pick \(1\), which simplifies to a positive, hence this interval is part of the solution. Do the same for \(3, 5\) and \(5, \infty\), pick 4 and 6 respectively. The inequality simplifies to a negative for \(4\) and positive for \(6\). Therefore, the intervals \(3, 5\) and \(5, \infty\) are not included in the solution set.

Step 4: Write the Solution in Interval Notation

Combine the solution intervals to write the solution in interval notation: \([0, 3] \)