Problem 23 - College Algebra

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 to the inequality \(-x^2 + x \geq 0\) in interval notation is \([0, 1]\).

Step 1: Rearrange inequality to Standard Form

Rearrange the inequality to standard form by adding \(x^{2}\) and subtracting \(x\) from both sides. This leads us to \(x^2-x \leq 0\)

Step 2: Identify Critical Points

Critical points are values of x which make the polynomial equation 0, so we solve \(x^2-x = 0\). This factorises to \(x(x-1) = 0\). Setting each factor to zero gives the critical points \(x=0\) and \(x=1\).

Step 3: Solve Inequality

Test a number from each interval between and beyond the critical points in the inequality. We have three intervals to consider: \((- \infinity, 0)\), \((0, 1)\), and \((1, \infinity)\). For the first interval, choose \(-1\), and the inequality \(1-(-1)=2\) is not less or equal to 0, so \((- \infinity, 0)\) is not part of the solution. For the second interval, choose \(0.5\), and the inequality \((0.5)^2 - 0.5 = -0.25\) is less or equal to 0, so \((0, 1)\) is part of the solution. For the last interval, choose \(2\), and the inequality \(2^2 - 2 = 2\) is not less or equal to 0, so \((1, \infinity)\) is not part of the solution.

Step 4: Write the Solution in Interval Notation

The only interval that satisfies the inequality is \((0, 1)\). The inequality includes \(0\) and \(1\) as it's 'less than or equal to' so the solution is \([0, 1]\).

Key Concepts

Real Number Line

The real number line is a powerful tool in mathematics that visually represents all real numbers in an orderly fashion. It extends infinitely in both positive and negative directions.
All real numbers are placed along this line, allowing us to visualize inequalities and critical points easily. When dealing with polynomial inequalities, the real number line helps us assess which intervals satisfy the inequality.
To solve an inequality such as \(-x^{2}+x \geq 0\), we plot the critical points on the real number line. This gives us a visual reference of possible solutions and helps us easily test values within different intervals.A common practice is to divide the number line around the critical points and test intervals to find which satisfy the inequality. This process shows how the solution set appears on this continuum, making abstract inequalities more tangible.

Interval Notation

Interval notation is a compact way of expressing the set of solutions to an inequality.It helps specify which parts of the real number line satisfy the given conditions. For example, the interval \[0, 1\] specifies all numbers between 0 and 1, including the endpoints.
When writing interval notation:

• Use square brackets, \[\,,\]\, to include endpoints, meaning the endpoints are part of the solution.
• Use parentheses, \((\,,\))\, to exclude endpoints, indicating they are not part of the solution.
• Combine intervals with the union symbol, \cup\, if multiple intervals satisfy the inequality.

Understanding interval notation is crucial for clearly communicating solutions. Once critical points are found and intervals are tested, interval notation answers the inequality question succinctly, like this: \[0, 1\].

Critical Points

Critical points determine key values of x that make a polynomial equal zero. They are fundamental in solving inequalities. To find them, you set the polynomial equal to zero and solve for x.For instance, in the inequality \(x^2-x \leq 0\), we simplify \(x(x-1) = 0\). The solutions \(x = 0\) and \(x = 1\) are our critical points.
These points divide the real number line into intervals: \((-\infty, 0)\), \(0, 1)\), and \(1, \infty)\).Testing values from each interval helps determine which satisfy the inequality. Critical points signal changes in inequality behavior, marking where intervals begin or end. Recognizing their role helps us efficiently solve polynomial inequalities and clearly define which intervals are included in the solution.