Problem 22 - 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. $$ 3 x^{2}-5 x \leq 0 $$

Step-by-Step Solution

Answer

The solution set to the inequality \(3x^2 - 5x \leq 0\) is \([0, 5/3]\).

Step 1: Factoring the inequality

To begin, factorise the inequality to make it easier to solve for x. We can take out common factors of x which gives: \(x(3x - 5) \leq 0\)

Step 2: Finding the critical points

The next step is to set each factor to zero and solve for x. This will give the critical points. Setting \(x = 0\) and \(3x - 5 = 0\), we find the critical points to be \(x = 0\) and \(x = 5/3\). Therefore, the critical points are \(0\) and \(5/3\).

Step 3: Testing intervals

With the critical points, we can divide the number line into intervals and then select a test point from each interval. The intervals are \(-\infty < x < 0\), \(0 < x < 5/3\), and \(5/3 < x < \infty\). For these intervals, choose test points, say \(-1\), \(1\), and \(2\), respectively. Substitute these points into the original inequality \(3x^2 - 5x \leq 0\). This will indicate whether the interval is part of the solution to the inequality.

Step 4: Writing the solution in interval notation

From the test, if the inequality holds true, it means the entire interval is a part of the solution set. If it doesn't, the interval is not included in the solution set. After testing the intervals for each test point within the original inequality, it has been determined that for \(x < 0\) and \(x > 5/3\) the inequality doesn't hold, but for \(0 \leq x \leq 5/3\), the inequality \(3x^2 - 5x \leq 0\) is satisfied by the values in this interval. In interval notation this is written as \([0, 5/3]\).