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. 3x² − 5x ≤ 0

Verified step by step guidance

1. Start by writing the inequality: \(3x^2 - 5x \leq 0\).
2. Factor the left-hand side expression: factor out the common term \(x\) to get \(x(3x - 5) \leq 0\).
3. Identify the critical points by setting each factor equal to zero: \(x = 0\) and \(3x - 5 = 0\), which gives \(x = \frac{5}{3}\).
4. Determine the sign of the product \(x(3x - 5)\) in the intervals defined by the critical points: \((-\infty, 0)\), \((0, \frac{5}{3})\), and \((\frac{5}{3}, \infty)\).
5. Use the sign analysis to find where the product is less than or equal to zero, then express the solution set in interval notation and graph it on the real number line.

Key Concepts

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (e.g., ≤, ≥, <, >). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.

Factoring Quadratic Expressions

Factoring is the process of rewriting a quadratic polynomial as a product of simpler binomials or monomials. For example, 3x² - 5x can be factored as x(3x - 5). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.

Interval Notation and Number Line Graphing

Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive, negative, or zero. Together, they help communicate the solution set clearly and precisely.