Question

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

Short answer

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

Step-by-step solution

Setting up the inequality

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

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.

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.

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 \([0,3]\).