Problem 23 - 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. $$ -x^{2}+x \geq 0 $$

Step-by-Step Solution

Answer

The solution in interval notation is \([- \infty, 1]\)

Step 1: Solving the inequality for zero

Solve the inequality equation for zero: \( -x^{2}+x \geq 0 \). It simplifies to \( x^{2} - x \leq 0 \)

Step 2: Factoring the polynomial

Factor the left side of the equation with common factor \( x \) to obtain \( x(x - 1) \leq 0 \)

Step 3: Finding the interval solutions

Set \( x \) and \( (x - 1) \) equal to zero to get \( x = 0, 1 \). Using a number line to demonstrate, the intervals become \( (-\infty, 0], [0, 1], (1, \infty) \)

Step 4: Testing the intervals

Choose a test point in each interval like, -1, 0.5, and 2. If test point makes inequality true, the interval is a solution. For this inequality, the intervals \((- \infty, 0] \) and \( [0, 1] \) make inequality true. If not, interval is not a solution.

Step 5: Express in interval notation

Combine the intervals \((- \infty, 0] \) and \( [0, 1] \) into a single notation \([- \infty, 1]\).