Solving $3x - 5 > 3x^2 - 9x + 4$ In Interval Notation A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the inequality $3x - 5 > 3x^2 - 9x + 4$. Inequalities, unlike equations, involve a range of solutions rather than discrete values. This particular inequality is a quadratic inequality, meaning it involves a polynomial of degree two. Solving such inequalities requires a combination of algebraic manipulation and an understanding of the behavior of quadratic functions. The solution will be expressed in interval notation, a standard way of representing continuous sets of numbers. Interval notation uses parentheses and brackets to indicate whether the endpoints of an interval are included or excluded. Mastering the techniques to solve quadratic inequalities is crucial for various mathematical applications, including calculus, optimization problems, and modeling real-world scenarios where constraints and boundaries are involved. This article aims to provide a comprehensive, step-by-step guide to solving this specific inequality, ensuring clarity and understanding for readers with varying levels of mathematical background. By the end of this guide, you should be able to confidently tackle similar quadratic inequalities and express their solutions in the correct interval notation. We will break down each step, providing explanations and insights to enhance your understanding of the underlying principles. So, let's embark on this mathematical journey and unravel the solution to this inequality, equipping you with valuable problem-solving skills.

Step 1: Rearrange the Inequality

The first crucial step in solving the inequality $3x - 5 > 3x^2 - 9x + 4$ is to rearrange it into a standard form that is easier to work with. The standard form for a quadratic inequality is $ax^2 + bx + c < 0$ or $ax^2 + bx + c > 0$. To achieve this, we need to move all terms to one side of the inequality, leaving zero on the other side. This process involves algebraic manipulation, ensuring we maintain the integrity of the inequality. The primary goal here is to consolidate all terms involving $x$ on one side, which will help us identify the coefficients and constants necessary for subsequent steps. By bringing all terms to one side, we also set the stage for factoring or using the quadratic formula, which are essential techniques for finding the critical points of the inequality. These critical points are the values of $x$ where the quadratic expression equals zero, and they play a pivotal role in determining the solution intervals. In essence, rearranging the inequality is not just a mechanical step; it's a strategic move that simplifies the problem and paves the way for a systematic solution. So, let’s proceed with this initial step and transform the inequality into its standard form, which will make the subsequent steps much clearer and more manageable. This rearrangement is the foundation upon which we will build our solution.

To start, we subtract $3x$ from both sides of the inequality:

$3x - 5 - 3x > 3x^2 - 9x + 4 - 3x$

This simplifies to:

$-5 > 3x^2 - 12x + 4$

Next, we add 5 to both sides:

$-5 + 5 > 3x^2 - 12x + 4 + 5$

Which gives us:

$0 > 3x^2 - 12x + 9$

Now, we can rewrite the inequality as:

$3x^2 - 12x + 9 < 0$

This form is now much easier to handle for the subsequent steps.

Step 2: Simplify the Quadratic Expression

Once we have rearranged the inequality into the standard form $3x^2 - 12x + 9 < 0$, the next step is to simplify the quadratic expression. Simplification often makes the expression easier to factor or solve using other methods. In this case, we observe that all the coefficients in the quadratic expression ($3x^2$, $-12x$, and $9$) are divisible by 3. Dividing the entire inequality by a positive constant does not change the direction of the inequality sign, which is a crucial property we can leverage to simplify the problem. Simplification not only reduces the magnitude of the numbers we are dealing with but also potentially reveals underlying structures or patterns that might have been obscured in the original expression. For instance, a simpler quadratic expression might be easier to factor, leading to a more straightforward determination of the critical points. Furthermore, simplification can reduce the chances of making arithmetic errors in the subsequent steps. Therefore, taking the time to simplify the quadratic expression is a worthwhile investment, as it streamlines the solution process and enhances accuracy. It’s a practical and efficient step that sets the stage for a more manageable analysis of the inequality.

We notice that all the coefficients are divisible by 3. So, we can divide the entire inequality by 3:

rac{3x^2 - 12x + 9}{3} < rac{0}{3}

This simplifies to:

$x^2 - 4x + 3 < 0$

This simplified quadratic expression is easier to factor and work with.

Step 3: Factor the Quadratic Expression

After simplifying the quadratic expression to $x^2 - 4x + 3 < 0$, the next critical step is to factor it. Factoring a quadratic expression involves rewriting it as a product of two binomials. This process is essential because it allows us to identify the values of $x$ that make the expression equal to zero, which are the critical points of the inequality. These critical points divide the number line into intervals, and the sign of the quadratic expression within each interval determines the solution set of the inequality. Factoring not only simplifies the process of finding these critical points but also provides valuable insights into the behavior of the quadratic function. For instance, the factors directly correspond to the roots of the equation, which are the x-intercepts of the parabola represented by the quadratic function. The ability to factor quadratic expressions is a fundamental skill in algebra and is crucial for solving a wide range of problems, including inequalities, equations, and optimization problems. In this particular case, factoring will transform the inequality into a form that allows us to easily identify the intervals where the expression is negative, thus solving the inequality. Let's proceed with factoring the simplified quadratic expression to unlock the critical points and determine the solution intervals.

To factor the quadratic expression $x^2 - 4x + 3$, we look for two numbers that multiply to 3 and add to -4. These numbers are -1 and -3. Therefore, we can rewrite the quadratic expression as:

$(x - 1)(x - 3) < 0$

This factored form is now much easier to analyze.

Step 4: Find the Critical Points

With the quadratic expression factored as $(x - 1)(x - 3) < 0$, the next crucial step is to find the critical points. Critical points are the values of $x$ that make the expression equal to zero. These points are essential because they divide the number line into intervals, within which the quadratic expression maintains a consistent sign (either positive or negative). Identifying these critical points is akin to finding the boundaries of the solution regions for the inequality. They represent the transition points where the expression changes its sign, and thus, they play a pivotal role in determining the solution set. The critical points are essentially the roots of the quadratic equation, and they can be easily found by setting each factor equal to zero and solving for $x$. These points serve as the foundation for constructing a sign chart, which is a visual tool used to analyze the sign of the expression across different intervals. Therefore, accurately determining the critical points is a fundamental step in solving quadratic inequalities, as it provides the framework for identifying the intervals that satisfy the inequality. Let's proceed to find these critical points and set the stage for constructing our sign chart.

To find the critical points, we set each factor equal to zero:

$x - 1 = 0$ and $x - 3 = 0$

Solving for $x$, we get:

$x = 1$ and $x = 3$

These critical points, $x = 1$ and $x = 3$, are the values where the expression $(x - 1)(x - 3)$ equals zero.

Step 5: Create a Sign Chart

After identifying the critical points, $x = 1$ and $x = 3$, the next essential step is to create a sign chart. A sign chart is a visual tool that helps us determine the intervals where the quadratic expression $(x - 1)(x - 3)$ is either positive or negative. This chart is constructed by placing the critical points on a number line, which divides the line into distinct intervals. Within each interval, the sign of each factor $(x - 1)$ and $(x - 3)$, as well as the sign of their product, remains constant. By analyzing the sign of each factor in each interval, we can deduce the sign of the entire expression. This is a powerful method because it provides a clear and organized way to visualize the behavior of the quadratic expression across the number line. The sign chart is particularly useful in solving inequalities because it allows us to easily identify the intervals that satisfy the given inequality condition, such as $(x - 1)(x - 3) < 0$. Constructing a sign chart is a systematic approach that reduces the likelihood of errors and ensures a comprehensive understanding of the solution. Let's proceed to create this chart and analyze the sign of the expression in each interval.

We create a number line and mark the critical points 1 and 3. This divides the number line into three intervals: $(- ext{∞}, 1)$, $(1, 3)$, and $(3, ext{∞})$.

In this sign chart:

• We choose a test value from each interval.
• We evaluate the sign of each factor $(x - 1)$ and $(x - 3)$ at the test value.
• We multiply the signs of the factors to determine the sign of the expression $(x - 1)(x - 3)$.

Step 6: Determine the Solution

Having constructed the sign chart, the next crucial step is to determine the solution to the inequality. The sign chart provides a clear visual representation of the intervals where the quadratic expression $(x - 1)(x - 3)$ is positive, negative, or zero. Since our original inequality is $3x^2 - 12x + 9 < 0$, which simplifies to $(x - 1)(x - 3) < 0$, we are looking for the intervals where the expression is negative. The sign chart directly indicates these intervals, allowing us to identify the range of $x$ values that satisfy the inequality. This is where the power of the sign chart becomes evident – it transforms a potentially complex inequality into a straightforward visual problem. By simply observing the chart, we can pinpoint the intervals where the product of the factors is negative. It is important to note whether the inequality is strict ($<$ or $>$) or includes equality ($ ext{≤}$ or $ ext{≥}$), as this determines whether the critical points themselves are included in the solution. In this case, since the inequality is strict ($<$), we will exclude the critical points from our solution. Let's now examine the sign chart to identify the intervals that satisfy our inequality and express the solution appropriately.

We are looking for the interval(s) where $(x - 1)(x - 3) < 0$. From the sign chart, we see that this occurs in the interval $(1, 3)$.

Since the inequality is strict (i.e., $<$), we do not include the critical points in the solution. Therefore, the solution is the open interval between 1 and 3.

Step 7: Write the Solution in Interval Notation

The final step in solving the inequality is to express the solution in interval notation. Interval notation is a standard way of representing a set of numbers that form an interval on the real number line. This notation uses parentheses and brackets to denote whether the endpoints of the interval are included or excluded. Parentheses, ( ), indicate that the endpoint is not included, which is used for strict inequalities ($<$ or $>$) and for intervals extending to infinity. Brackets, [ ], indicate that the endpoint is included in the interval, which is used for inequalities that include equality ($ ext{≤}$ or $ ext{≥}$). Interval notation provides a concise and unambiguous way to represent solution sets, especially for inequalities where the solution is a continuous range of values. This notation is widely used in mathematics and is essential for communicating solutions effectively. In our case, since we have determined that the solution lies between 1 and 3, and the inequality is strict, we will use parentheses to indicate that neither 1 nor 3 is included in the solution set. Let's express our solution in the appropriate interval notation, completing the solution to the inequality.

The solution to the inequality $3x - 5 > 3x^2 - 9x + 4$ is the interval $(1, 3)$. This means that all values of $x$ between 1 and 3 (excluding 1 and 3) satisfy the inequality.

Conclusion

In conclusion, we have successfully solved the inequality $3x - 5 > 3x^2 - 9x + 4$ and expressed the solution in interval notation. The process involved several key steps, each building upon the previous one to arrive at the final answer. First, we rearranged the inequality into the standard quadratic form, making it easier to manipulate. Next, we simplified the quadratic expression by dividing by a common factor, which reduced the complexity of the problem. Factoring the quadratic expression was a critical step, as it allowed us to identify the critical points, which are the roots of the equation. These critical points served as the boundaries for intervals on the number line. We then constructed a sign chart, a powerful tool for visualizing the sign of the quadratic expression in each interval. By analyzing the sign chart, we determined the interval(s) where the inequality holds true. Finally, we expressed the solution in interval notation, providing a clear and concise representation of the solution set. This step-by-step approach not only provides the solution to this specific inequality but also equips you with a systematic method for solving similar problems. Understanding these techniques is essential for further studies in mathematics and related fields. With practice, you can confidently tackle a wide range of inequalities and express their solutions effectively. The journey of solving this inequality has reinforced the importance of algebraic manipulation, factoring, and the use of visual aids like sign charts in mathematical problem-solving.