Questions: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. x^2-3x >= 0 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.

$Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. x^2-3x >= 0 Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is . (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.$

Transcript text: Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. \[ x^{2}-3 x \geq 0 \] Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\square$ . (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.

Solution

Solution Steps

Step 1: Solve the inequality

To solve the inequality $x^2 - 3x \geq 0$, we first find the roots of the equation $x^2 - 3x = 0$.

Step 2: Find the roots

Factor the quadratic equation: \[ x(x - 3) = 0 \]

The roots are: \[ x = 0 \quad \text{and} \quad x = 3 \]

Step 3: Determine the intervals

The roots divide the number line into intervals: $(-\infty, 0)$, $[0, 3]$, and $(3, \infty)$.

Step 4: Test the intervals

For $x \in (-\infty, 0)$, choose $x = -1$: \[ (-1)^2 - 3(-1) = 1 + 3 = 4 \geq 0 \]
For $x \in (0, 3)$, choose $x = 1$: \[ 1^2 - 3(1) = 1 - 3 = -2 \not\geq 0 \]
For $x \in (3, \infty)$, choose $x = 4$: \[ 4^2 - 3(4) = 16 - 12 = 4 \geq 0 \]