How to Solve a Polynomial Inequality

Oct 13, 2025 by leolavanderia.com.br 83 views

**Solve the Polynomial Inequality and Graph the Solution Set on a Real Number Line** ===========================================================

Introduction

In this article, we will learn how to solve a polynomial inequality and graph the solution set on a real number line. We will also express the solution set in interval notation. The given inequality is $5x^2 - 6x > 0$ . We will use algebraic methods to solve the inequality and then graph the solution set on a real number line.

What is a Polynomial Inequality?

A polynomial inequality is an inequality that contains a polynomial expression. It is an expression that involves variables and coefficients, and the variables are raised to non-negative integer powers. In this case, the polynomial expression is $5x^2 - 6x$ .

How to Solve a Polynomial Inequality

To solve a polynomial inequality, we need to follow these steps:

Factor the polynomial expression: If the polynomial expression can be factored, we need to factor it.
Find the critical points: We need to find the values of x that make the polynomial expression equal to zero.
Test the intervals: We need to test the intervals between the critical points to see which intervals satisfy the inequality.
Graph the solution set: We need to graph the solution set on a real number line.

Solving the Inequality

Let's solve the inequality $5x^2 - 6x > 0$ .

Step 1: Factor the Polynomial Expression

The polynomial expression $5x^2 - 6x$ can be factored as $5x(x - \frac{6}{5})$ .

Step 2: Find the Critical Points

The critical points are the values of x that make the polynomial expression equal to zero. In this case, the critical points are $x = 0$ and $x = \frac{6}{5}$ .

Step 3: Test the Intervals

We need to test the intervals between the critical points to see which intervals satisfy the inequality. The intervals are $(-\infty, 0)$ , $(0, \frac{6}{5})$ , and $(\frac{6}{5}, \infty)$ .

Step 4: Graph the Solution Set

We need to graph the solution set on a real number line. The solution set is the set of all values of x that satisfy the inequality.

Q&A

Q: What is a polynomial inequality? A: A polynomial inequality is an inequality that contains a polynomial expression.

Q: How do I solve a polynomial inequality? A: To solve a polynomial inequality, you need to factor the polynomial expression, find the critical points, test the intervals, and graph the solution set.

Q: What are the critical points? A: The critical points are the values of x that make the polynomial expression equal to zero.

Q: How do I test the intervals? A: You need to test the intervals between the critical points to see which intervals satisfy the inequality.

Q: What is the solution set? A: The solution set is the set of all values of x that satisfy the inequality.

Q: How do I graph the solution set? A: You need to graph the solution set on a real number line.

Conclusion

In this article, we learned how to solve a polynomial inequality and graph the solution set on a real number line. We also expressed the solution set in interval notation. The given inequality was $5x^2 - 6x > 0$ . We used algebraic methods to solve the inequality and then graphed the solution set on a real number line.

Final Answer

The final answer is $\boxed{(0, \frac{6}{5})}$ .

Note

The solution set is expressed in interval notation. The interval notation is a way of expressing a set of numbers using parentheses and brackets. In this case, the solution set is expressed as $(0, \frac{6}{5})$ , which means that the solution set includes all values of x that are greater than 0 and less than $\frac{6}{5}$ .