Problem 54 - College Algebra and Calculus: An Applied Approach

Question

Solve the inequality and write the solution set in interval notation. \(x^{4}(x-3) \leq 0\)

Step-by-Step Solution

Answer

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

Step 1: Setting up the inequality

The given inequality is \(x^{4}(x-3) \leq 0\). Set the expression equal to zero; this gives \(x^{4}(x-3) = 0\).

Step 2: Finding the zeroes or root points

Solve for x by setting each part of the equation = 0. We get the values of x as 0 and 3. These are the points where the expression on the left side changes its sign.

Step 3: Test the intervals

The intervals we have are \(-\infty ,0\), \(0,3\) and \(3, \infty\). Now choose a test point from each interval, substitute it into the original inequality equation and check the sign. We take -1 for the first interval, 1 for the second interval, and 4 for the third interval. After substitution, we find that the signs for the first and third intervals are negative and for the second interval, it is positive.

Step 4: Write the solution in interval notation

Since the inequality allows for the expression to be equal to 0, we know that the end points which are 0 and 3 are included in the solution. Thus, the solution to the inequality \(x^{4}(x-3) \leq 0\) in interval notation is \([0,3]\).