Solving Polynomial Inequalities A Step By Step Guide To $3x^2 - 2x < 5$

Polynomial inequalities are mathematical expressions that compare a polynomial to another value using inequality symbols such as <, >, ≤, or ≥. Solving polynomial inequalities involves finding the range of values for the variable that satisfy the inequality. This guide provides a detailed, step-by-step approach to solving the polynomial inequality $3x^2 - 2x < 5$, graphing the solution set on a real number line, and expressing the solution set in interval notation. Mastering these techniques is crucial for various applications in mathematics, physics, and engineering. Before diving into the specifics of this inequality, it's essential to understand the foundational concepts that underpin polynomial inequalities. Polynomial inequalities, unlike polynomial equations, do not have a fixed set of solutions. Instead, they have a range of values that satisfy the given condition. This makes the solution process slightly more intricate, often involving graphical representations and interval notations to accurately describe the solution set. The process typically begins with transforming the inequality into a standard form, which involves setting one side of the inequality to zero. This step is critical as it allows us to identify the critical points, which are the values of the variable that make the polynomial equal to zero. These critical points serve as boundaries that divide the number line into intervals, each of which needs to be tested to determine if it satisfies the original inequality. Understanding this initial setup is key to successfully navigating the complexities of solving polynomial inequalities. The beauty of polynomial inequalities lies in their ability to model real-world scenarios where quantities are not exact but fall within a certain range. For instance, in physics, one might use a polynomial inequality to describe the range of possible velocities for a projectile given certain constraints. Similarly, in economics, these inequalities can help define the range of production levels that maximize profit. The ability to solve these inequalities, therefore, transcends the classroom and becomes a vital tool in various practical applications. As we proceed with solving the specific inequality $3x^2 - 2x < 5$, remember that the underlying principles discussed here apply broadly to all polynomial inequalities. By grasping these fundamentals, you'll be well-equipped to tackle a wide array of problems, enhancing your problem-solving skills and deepening your understanding of mathematical concepts.

Step 1: Rewrite the Inequality

The first crucial step in solving polynomial inequalities is to rewrite the inequality so that one side is zero. This transformation allows us to easily identify the critical points where the polynomial equals zero, which are essential for determining the solution intervals. In our case, we start with the inequality:

$$3x^2 - 2x < 5$$

To set one side to zero, we subtract 5 from both sides of the inequality:

$$3x^2 - 2x - 5 < 0$$

This rewritten form sets the stage for the next steps in the solution process. The importance of this step cannot be overstated. By bringing all terms to one side, we create a standard form that makes it easier to analyze the polynomial. The polynomial expression $3x^2 - 2x - 5$ now represents a function whose sign we are interested in—specifically, where it is less than zero. This standard form simplifies the task of finding the intervals where the inequality holds true. Furthermore, this method is universally applicable to all polynomial inequalities, regardless of their degree or complexity. Whether you are dealing with a quadratic, cubic, or higher-degree polynomial, the initial step of setting the inequality to zero is always the same. It is a fundamental technique that streamlines the solution process and allows for a consistent approach to solving a wide range of problems. Understanding this step thoroughly will empower you to confidently tackle various polynomial inequalities. Additionally, rewriting the inequality in this manner also sets the foundation for graphical analysis. The polynomial $3x^2 - 2x - 5$ can be visualized as a parabola, and finding where it is less than zero corresponds to identifying the regions where the parabola lies below the x-axis. This graphical perspective provides an alternative way to understand the solution and can be particularly helpful for students who are visual learners. In essence, rewriting the inequality to have zero on one side is not just a mechanical step; it's a strategic move that simplifies the algebraic manipulations and opens the door to both analytical and graphical methods of solving polynomial inequalities. This foundational step is a cornerstone of the entire process, and mastering it is key to successfully navigating more complex problems.

Step 2: Factor the Quadratic Expression

Once we have the inequality in the form $3x^2 - 2x - 5 < 0$, the next step is to factor the quadratic expression. Factoring allows us to find the roots of the polynomial, which are the critical points that divide the number line into intervals. In this case, we need to factor the quadratic $3x^2 - 2x - 5$. To factor this quadratic, we look for two binomials that multiply to give us the original quadratic expression. We can rewrite the middle term (-2x) as a sum of two terms such that the coefficients multiply to the product of the leading coefficient (3) and the constant term (-5), which is -15. The two numbers that satisfy this condition are -5 and 3. So, we can rewrite the quadratic expression as follows:

$$3x^2 - 5x + 3x - 5$$

Now, we can factor by grouping:

$$x(3x - 5) + 1(3x - 5)$$

$$(3x - 5)(x + 1)$$

Thus, the factored form of the quadratic expression is $(3x - 5)(x + 1)$. Factoring the quadratic expression is a crucial step because it transforms the inequality into a product of linear factors, making it easier to determine the intervals where the expression is positive or negative. The roots of the polynomial, which are the values of x that make the polynomial equal to zero, are found by setting each factor equal to zero and solving for x. These roots serve as the critical points that divide the number line into regions where the inequality may or may not hold true. The technique of factoring, particularly for quadratic expressions, is a fundamental skill in algebra. It is not only essential for solving inequalities but also plays a vital role in simplifying algebraic expressions, solving equations, and graphing functions. Mastering factoring techniques will significantly enhance your ability to tackle a wide range of mathematical problems. In this specific case, factoring the quadratic expression allowed us to identify the values of x that make the polynomial equal to zero, which are the boundaries of the intervals we need to test. These critical points are the solutions to the equations $3x - 5 = 0$ and $x + 1 = 0$, which we will determine in the next step. Furthermore, the factored form of the quadratic expression provides valuable insight into the behavior of the corresponding parabola. The factors $(3x - 5)$ and $(x + 1)$ reveal the x-intercepts of the parabola, which are the points where the graph crosses the x-axis. This connection between the algebraic factored form and the graphical representation is a powerful tool for understanding and solving polynomial inequalities. Understanding the relationship between factoring, roots, and the graph of a polynomial is key to developing a comprehensive understanding of polynomial functions and their applications.

Step 3: Find the Critical Points

After factoring the quadratic expression, the next crucial step is to find the critical points. Critical points are the values of $x$ that make the polynomial equal to zero. These points are vital because they divide the number line into intervals, each of which we will test to determine if it satisfies the inequality. From the factored form $(3x - 5)(x + 1)$, we set each factor equal to zero and solve for $x$:

1. $3x - 5 = 0$

   $$3x = 5$$

   x = rac{5}{3}

2. $x + 1 = 0$

   $$x = -1$$

Thus, the critical points are $x = -1$ and x = rac{5}{3}. These critical points are the cornerstone of the solution process. They are the values of $x$ where the polynomial changes sign, transitioning from positive to negative or vice versa. By identifying these points, we can create intervals on the number line that we will test to determine where the inequality $3x^2 - 2x - 5 < 0$ holds true. The concept of critical points is not limited to quadratic inequalities; it extends to polynomial inequalities of any degree. For a polynomial of degree $n$, there can be up to $n$ real roots, each of which serves as a critical point. These points are essential for creating a sign chart, a visual tool that helps determine the intervals where the polynomial is positive, negative, or zero. Understanding how to find and use critical points is a fundamental skill in solving polynomial inequalities. Furthermore, the critical points have a direct graphical interpretation. They represent the x-intercepts of the graph of the polynomial function. In other words, they are the points where the graph crosses the x-axis. This connection between the algebraic solution (critical points) and the graphical representation (x-intercepts) provides a deeper understanding of the behavior of polynomial functions. Visualizing the graph can often help confirm the algebraic solution and provide additional insight into the problem. In the context of our specific inequality, the critical points $x = -1$ and x = rac{5}{3} divide the number line into three intervals: $(-\infty, -1)$, (-1, rac{5}{3}), and (rac{5}{3}, \infty). We will test each of these intervals in the next step to determine which ones satisfy the inequality $3x^2 - 2x - 5 < 0$. By mastering the technique of finding critical points, you'll be well-equipped to tackle a wide range of polynomial inequalities and gain a deeper understanding of their solutions.

Step 4: Create a Sign Chart and Test Intervals

Once we have identified the critical points, the next crucial step is to create a sign chart and test intervals. This method helps us determine the intervals on the number line where the inequality $3x^2 - 2x - 5 < 0$ holds true. The critical points, which we found to be $x = -1$ and x = rac{5}{3}, divide the number line into three intervals: $(-\infty, -1)$, $(-1, \frac{5}{3})$, and $(\frac{5}{3}, \infty)$. To determine the sign of the polynomial $(3x - 5)(x + 1)$ in each interval, we choose a test value within each interval and substitute it into the factored expression. The sign of the result will tell us whether the polynomial is positive or negative in that interval.

1. Interval $(-\infty, -1)$:

   Choose a test value, say $x = -2$. Substitute it into the factored expression:

   $$(3(-2) - 5)((-2) + 1) = (-6 - 5)(-1) = (-11)(-1) = 11$$

   Since the result is positive, the polynomial is positive in this interval.

2. Interval $(-1, \frac{5}{3})$:

   Choose a test value, say $x = 0$. Substitute it into the factored expression:

   $$(3(0) - 5)((0) + 1) = (-5)(1) = -5$$

   Since the result is negative, the polynomial is negative in this interval.

3. Interval $(\frac{5}{3}, \infty)$:

   Choose a test value, say $x = 2$. Substitute it into the factored expression:

   $$(3(2) - 5)((2) + 1) = (6 - 5)(3) = (1)(3) = 3$$

   Since the result is positive, the polynomial is positive in this interval.

Now, we create a sign chart to summarize our findings:

The sign chart is a powerful visual tool that summarizes the sign of the polynomial in each interval. It clearly shows where the polynomial is positive, negative, or zero. This information is essential for determining the solution set of the inequality. The method of testing intervals is a general technique that applies to all polynomial inequalities. By choosing a test value within each interval and substituting it into the factored expression, we can efficiently determine the sign of the polynomial in that interval. This technique simplifies the process of solving inequalities and provides a systematic approach to finding the solution set. Furthermore, the sign chart helps to visualize the solution set on the number line. The intervals where the polynomial satisfies the inequality (in this case, less than zero) are clearly indicated on the chart. This visual representation makes it easier to understand the solution and to express it in interval notation. In the context of our specific inequality, the sign chart shows that the polynomial $(3x - 5)(x + 1)$ is negative in the interval $(-1, \frac{5}{3})$. This means that the inequality $3x^2 - 2x - 5 < 0$ is satisfied in this interval. In the next step, we will use this information to express the solution set in interval notation and graph it on the number line. By mastering the technique of creating sign charts and testing intervals, you'll be well-equipped to solve a wide range of polynomial inequalities and gain a deeper understanding of their solutions.

Step 5: Determine the Solution Set

With the sign chart complete, we can now determine the solution set for the inequality $3x^2 - 2x - 5 < 0$. The sign chart indicates the intervals where the polynomial $(3x - 5)(x + 1)$ is negative. From our sign chart, we found that the polynomial is negative in the interval $(-1, \frac{5}{3})$. Since the inequality is strictly less than zero (i.e., $<$), we do not include the critical points $x = -1$ and $x = \frac{5}{3}$ in the solution set. Therefore, the solution set in interval notation is:

$$(-1, \frac{5}{3})$$

This interval represents all the values of $x$ that satisfy the inequality $3x^2 - 2x < 5$. To further illustrate the solution, we can graph it on a real number line. We draw an open circle at $x = -1$ and $x = \frac{5}{3}$ to indicate that these points are not included in the solution set. Then, we shade the region between these two points to represent the interval $(-1, \frac{5}{3})$. The solution set is the range of values for x that make the inequality true. In this case, the solution is all real numbers between -1 and 5/3, not including the endpoints themselves. This is because the inequality is strictly less than (<), meaning the polynomial must be negative, not zero. If the inequality had been less than or equal to (≤), we would have included the endpoints in the solution set, using closed circles on the number line and square brackets in the interval notation. Understanding the difference between strict and non-strict inequalities is crucial for accurately determining the solution set. Non-strict inequalities (≤ or ≥) include the endpoints, while strict inequalities (< or >) do not. This distinction directly affects how we represent the solution set in both interval notation and on the number line. Furthermore, the solution set can be visualized graphically. The inequality $3x^2 - 2x - 5 < 0$ corresponds to the region where the parabola $y = 3x^2 - 2x - 5$ lies below the x-axis. The interval $(-1, \frac{5}{3})$ represents the x-values for which the parabola is below the x-axis. This graphical interpretation provides a powerful visual confirmation of the algebraic solution. In summary, determining the solution set involves interpreting the sign chart, considering the type of inequality (strict or non-strict), and representing the solution in interval notation and on the number line. This process provides a comprehensive understanding of the range of values that satisfy the given inequality. By mastering these techniques, you'll be well-equipped to solve a wide range of polynomial inequalities and effectively communicate their solutions.

Step 6: Graph the Solution Set on a Real Number Line

To visually represent the solution set, we graph it on a real number line. This graphical representation provides a clear and intuitive understanding of the values of $x$ that satisfy the inequality $3x^2 - 2x < 5$. We found that the solution set is the interval $(-1, \frac{5}{3})$. To graph this interval on a number line, we follow these steps:

1. Draw a real number line.
2. Locate the critical points $x = -1$ and $x = \frac{5}{3}$ on the number line.
3. Since the inequality is strictly less than (i.e., $<$), we use open circles at $x = -1$ and $x = \frac{5}{3}$ to indicate that these points are not included in the solution set.
4. Shade the region between $-1$ and $\frac{5}{3}$ to represent the interval $(-1, \frac{5}{3})$.

The graph on the real number line visually confirms that the solution set includes all real numbers between $-1$ and $\frac{5}{3}$, excluding the endpoints themselves. Graphing the solution set on a real number line is an essential step in solving inequalities. It provides a visual representation of the solution, making it easier to understand and interpret. The number line clearly shows the range of values that satisfy the inequality, as well as the values that do not. This visual aid is particularly helpful for students who are visual learners. Furthermore, the graph on the number line directly corresponds to the interval notation of the solution set. The open circles at the critical points indicate that the endpoints are not included, while the shaded region represents the interval between the endpoints. This connection between the graphical representation and the interval notation reinforces the understanding of the solution. The use of open and closed circles on the number line is a standard convention for representing strict and non-strict inequalities, respectively. Open circles indicate that the endpoint is not included, while closed circles indicate that the endpoint is included. This convention ensures that the graph accurately represents the solution set. In the context of our specific inequality, the graph clearly shows that the solution set consists of all real numbers between -1 and 5/3, not including -1 and 5/3 themselves. This graphical representation provides a comprehensive understanding of the solution and complements the algebraic solution obtained earlier. By mastering the technique of graphing solution sets on a real number line, you'll be well-equipped to visually represent the solutions of inequalities and gain a deeper understanding of their properties.

Step 7: Express the Solution Set in Interval Notation

The final step in solving the polynomial inequality is to express the solution set in interval notation. Interval notation is a concise way to represent a set of real numbers using intervals. We have already determined that the solution set for the inequality $3x^2 - 2x < 5$ is the interval between $-1$ and $\frac{5}{3}$, excluding the endpoints. In interval notation, this is written as:

$$(-1, \frac{5}{3})$$

In interval notation, parentheses () are used to indicate that the endpoints are not included in the interval, while square brackets [] are used to indicate that the endpoints are included. Since our inequality is strictly less than, we use parentheses to exclude the critical points $x = -1$ and $x = \frac{5}{3}$. Expressing the solution set in interval notation is a standard practice in mathematics. It provides a clear and concise way to represent a range of values. Interval notation is particularly useful when dealing with inequalities, as it accurately captures the set of all real numbers that satisfy the inequality. The use of parentheses and square brackets in interval notation is a fundamental concept that must be understood to correctly represent solution sets. Parentheses indicate that the endpoint is not included, while square brackets indicate that the endpoint is included. This distinction is crucial for accurately representing the solution of an inequality. Furthermore, interval notation can be used to represent various types of intervals, including open intervals (endpoints not included), closed intervals (endpoints included), half-open intervals (one endpoint included), and infinite intervals (extending to infinity). The ability to express solution sets in interval notation is a valuable skill that is used extensively in calculus, analysis, and other advanced mathematical topics. In the context of our specific inequality, the interval notation $(-1, \frac{5}{3})$ clearly represents the set of all real numbers between -1 and 5/3, excluding -1 and 5/3 themselves. This concise notation accurately captures the solution set and is easily understood by mathematicians and students alike. By mastering the technique of expressing solution sets in interval notation, you'll be well-equipped to communicate mathematical solutions effectively and gain a deeper understanding of the properties of intervals.

Summary: Solution to $3x^2 - 2x < 5$

In summary, to solve the polynomial inequality $3x^2 - 2x < 5$, we followed these steps:

1. Rewrite the inequality: $3x^2 - 2x - 5 < 0$
2. Factor the quadratic expression: $(3x - 5)(x + 1) < 0$
3. Find the critical points: $x = -1$ and $x = \frac{5}{3}$
4. Create a sign chart and test intervals: Determine the sign of the polynomial in each interval $(-\infty, -1)$, $(-1, \frac{5}{3})$, and $(\frac{5}{3}, \infty)$.
5. Determine the solution set: The inequality is satisfied in the interval $(-1, \frac{5}{3})$.
6. Graph the solution set on a real number line: Shade the region between $-1$ and $\frac{5}{3}$ with open circles at the endpoints.
7. Express the solution set in interval notation: $(-1, \frac{5}{3})$

This step-by-step approach provides a comprehensive solution to the inequality. By following these steps, you can confidently solve a wide range of polynomial inequalities. The process of solving polynomial inequalities involves a combination of algebraic techniques, graphical representations, and logical reasoning. Each step in the solution process builds upon the previous one, leading to a clear and accurate solution. Mastering these steps is essential for success in algebra and calculus. Furthermore, the ability to solve polynomial inequalities has practical applications in various fields, including engineering, physics, and economics. These inequalities can be used to model real-world scenarios and to find optimal solutions to problems. The techniques discussed in this guide are not limited to quadratic inequalities; they can be extended to polynomial inequalities of higher degrees. The key is to factor the polynomial, find the critical points, create a sign chart, and test the intervals. This systematic approach provides a general method for solving any polynomial inequality. In conclusion, solving polynomial inequalities is a fundamental skill in mathematics. By understanding the steps involved and practicing the techniques, you can develop a strong foundation in algebra and gain valuable problem-solving skills. The solution to the inequality $3x^2 - 2x < 5$ is the interval $(-1, \frac{5}{3})$, which represents all real numbers between -1 and 5/3, not including the endpoints. This solution can be visualized on a number line and expressed concisely in interval notation.

By following this comprehensive guide, you can confidently solve polynomial inequalities and express the solution set in interval notation.