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. x³≤4x²

Verified step by step guidance

1. Rewrite the inequality $x^3\le 4x^2$ by bringing all terms to one side to set the inequality to zero: $x^3-4x^2\le \; 0$.
2. Factor the expression on the left side. First, factor out the greatest common factor $x^2$: $x^2(x\; -\; 4)\; \le \; 0$.
3. Identify the critical points by setting each factor equal to zero: $x^2=\; 0$ gives $x\; =\; 0$, and $x\; -\; 4\; =\; 0$ gives $x\; =\; 4$. These points divide the real number line into intervals to test.
4. Test values from each interval determined by the critical points () in the factored inequality $x^2(x\; -\; 4)\; \le \; 0$ to determine where the inequality holds true.
5. Express the solution set by including the points where the expression equals zero (since the inequality is ≤ 0) and the intervals where the inequality is satisfied. Then write 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 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 Polynomials

Factoring is the process of rewriting a polynomial as a product of simpler polynomials. It helps identify the roots or zeros of the polynomial, which are critical points for determining where the polynomial changes sign in inequality problems.

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 satisfies the inequality, using open or closed dots to indicate whether endpoints are included.