Problem 24 - 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. $$ -x^{2}+2 x \geq 0 $$

Step-by-Step Solution

Answer

The solution to the polynomial inequality is \(x \in [0, 2]\).

Step 1: Rewrite the Inequality

The inequality \(-x^{2}+2 x \geq 0\) can be rewritten as \(x^2 - 2x \leq 0\).

Step 2: Factoring the Polynomial

Now, factor the polynomial on the left-hand side. The equation becomes \(x(x - 2) \leq 0\). This provides the zeros of the polynomial, which are x = 0 and x = 2.

Step 3: Test the Intervals

The real numbers are divided into three intervals by the zeros: \((-∞, 0)\), \((0, 2)\), and \((2, ∞)\). Pick a test point from each interval: let's say -1, 1, and 3 and plug these points into the factored inequality. If the inequality is true for the test point, then it's true for the entire interval. Our test returns that the inequality is true only for the interval between 0 and 2.

Step 4: Graphing the Solution

Plot on a number line with a closed circle at 0 and 2 to indicate that the endpoints are included in the solution set, as the original inequality is 'less than or equal to' not 'less than'. Fill in the line segment between the endpoints.

Step 5: Express the Solution in Interval Notation

The solution represented on the number line can be written in interval notation as \([0, 2]\). The square brackets include the endpoints.