Solve The Inequality -6a - 2 < -74 And Describe The Solution Graph.

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Inequalities extend the concept of equations by introducing symbols that represent greater than (>), less than (<), greater than or equal to (≥), and less than or equal to (≤). Mastering the art of solving inequalities is fundamental for anyone delving into algebra, calculus, and various other mathematical disciplines. This guide aims to provide a comprehensive understanding of solving linear inequalities, focusing on a step-by-step approach and graphical representation of solutions. We will dissect the process of isolating the variable, handling negative coefficients, and interpreting the solution set on a number line. Let's delve into the intricacies of inequalities and equip ourselves with the tools to tackle them effectively.

Understanding Linear Inequalities

Before we dive into the process of solving linear inequalities, let's first establish a solid understanding of what they are and how they differ from linear equations. Linear inequalities, at their core, are mathematical statements that compare two expressions using inequality symbols. Unlike linear equations, which seek to find specific values that make the expressions equal, linear inequalities identify a range of values that satisfy the given relationship. This distinction is crucial because it leads to solutions that are not single points but rather intervals on the number line. For instance, the inequality x > 3 signifies that x can take on any value greater than 3, representing an infinite set of solutions. This concept of a solution set is a hallmark of inequalities and sets them apart from equations. Grasping this fundamental difference is paramount as we move forward in solving and interpreting linear inequalities. It's also essential to recognize that the properties of equality, such as addition, subtraction, multiplication, and division, have analogous counterparts in inequalities, with a critical exception: multiplying or dividing by a negative number reverses the direction of the inequality. This seemingly small detail is a common pitfall for students and underscores the importance of meticulous attention to detail when manipulating inequalities. The solutions to linear inequalities can be visualized on a number line, providing a clear and intuitive representation of the range of possible values. This graphical representation is a powerful tool for understanding the implications of the inequality and forms a cornerstone of the solution process.

Step-by-Step Solution: -6a - 2 < -74

To effectively solve the inequality ${-6a - 2 < -74,}$ we will embark on a methodical journey, following a series of algebraic steps designed to isolate the variable a. This process mirrors that of solving linear equations, but with a critical caveat: the direction of the inequality sign must be carefully considered, especially when multiplying or dividing by negative numbers. Our initial goal is to isolate the term containing the variable. We achieve this by adding 2 to both sides of the inequality. This operation maintains the balance of the inequality while strategically moving us closer to isolating a. The addition property of inequality allows us to add the same value to both sides without altering the solution set. Next, we focus on isolating a itself. This requires us to eliminate the coefficient -6. To do so, we divide both sides of the inequality by -6. Here's where the crucial rule comes into play: dividing by a negative number necessitates flipping the direction of the inequality sign. This is a fundamental principle in inequality manipulation, ensuring the solution set remains accurate. The act of flipping the sign is not merely a mechanical step but a reflection of the number line's structure. Dividing by a negative number effectively mirrors the interval across the origin, thus reversing the order of the values. Once we've performed the division and flipped the inequality sign, we arrive at the solution for a. This solution represents a range of values, not a single point, and is best understood in the context of the number line. It's crucial to interpret the solution correctly, understanding whether the endpoint is included or excluded from the solution set. This is often visually represented using open or closed circles on the number line, a detail we'll explore further in the graphical representation section. By meticulously following these steps, we can confidently navigate the intricacies of linear inequalities and arrive at accurate solutions.

1. Isolate the term with the variable: Add 2 to both sides of the inequality: ${-6a - 2 + 2 < -74 + 2}$ ${-6a < -72}$

2. Isolate the variable: Divide both sides by -6. Remember to flip the inequality sign since we are dividing by a negative number: ${\frac{-6a}{-6} > \frac{-72}{-6}}$ ${a > 12}$

Solution: a > 12

Our algebraic journey has culminated in a clear and concise solution: a > 12. This solution is more than just a mathematical expression; it's a gateway to understanding the range of values that satisfy the original inequality. The solution a > 12 signifies that any value of a strictly greater than 12 will make the inequality ${-6a - 2 < -74}$ true. This underscores the fundamental difference between solving equations and inequalities: equations typically yield a finite set of solutions, while inequalities unveil an infinite spectrum of possibilities. The solution set a > 12 stretches endlessly along the number line, encompassing all real numbers exceeding 12. To fully grasp the implications of this solution, it's crucial to consider the endpoint, 12. The inequality symbol '>' indicates that 12 itself is not included in the solution set. This subtle yet significant detail dictates how we represent the solution graphically. The exclusion of 12 is often denoted by an open circle on the number line, a visual cue that distinguishes it from closed circles, which signify inclusion. The solution a > 12 is a testament to the power of algebraic manipulation in unveiling hidden relationships between variables. It's a concise statement that encapsulates a wealth of information about the possible values of a. However, the true essence of this solution is best appreciated when visualized on a number line, where the infinite nature of the solution set becomes strikingly apparent. This graphical representation not only enhances our understanding but also serves as a bridge to more complex mathematical concepts, such as interval notation and set theory.

The solution to the inequality is: ${a > 12}$

Graphing the Solution on a Number Line

Visualizing the solution to an inequality on a number line is a powerful technique that transforms an abstract algebraic concept into a tangible representation. The number line becomes a canvas upon which we paint the range of values that satisfy the inequality, providing an immediate and intuitive understanding of the solution set. For the inequality a > 12, the graphical representation is particularly insightful. First, we locate the critical value, 12, on the number line. This point serves as the boundary of our solution set. The next step is to determine whether 12 itself is included in the solution. Since our inequality uses the '>' symbol (greater than), 12 is excluded. This exclusion is visually represented by drawing an open circle at the point 12 on the number line. An open circle acts as a clear indicator that 12 is not part of the solution set, emphasizing the strict inequality. Now, we must represent all values greater than 12. This is achieved by shading the portion of the number line to the right of 12. The shading extends indefinitely, signifying that the solution set encompasses all real numbers greater than 12, stretching towards positive infinity. The arrowhead at the end of the shaded region further reinforces this infinite extension. The combination of the open circle at 12 and the shaded region to its right creates a complete visual representation of the solution a > 12. This graphical depiction is not merely a cosmetic addition; it's a fundamental tool for understanding the nature of inequalities. It allows us to quickly grasp the range of solutions and serves as a bridge between the algebraic solution and its real-world implications. Furthermore, the number line representation lays the groundwork for understanding more complex concepts such as interval notation, which provides a concise way to express solution sets.

• Circle Type: An open circle is used at 12 because the solution is strictly greater than 12.
• Shading Direction: The number line is shaded to the right of 12, indicating all values greater than 12 are solutions.

Conclusion

In this comprehensive exploration, we have meticulously dissected the process of solving linear inequalities, focusing on the specific example of ${-6a - 2 < -74.}$ We embarked on a step-by-step algebraic journey, mastering the art of isolating the variable while adhering to the crucial rule of flipping the inequality sign when multiplying or dividing by a negative number. This meticulous manipulation led us to the solution a > 12, a statement that encapsulates an infinite range of values. However, the true power of our understanding was unleashed when we translated this algebraic solution into a graphical representation on the number line. The number line served as a visual canvas, allowing us to paint the solution set with clarity and precision. The open circle at 12 served as a stark reminder of the strict inequality, while the shaded region extending to the right conveyed the infinite nature of the solution. This graphical representation not only solidified our understanding but also provided a foundation for more advanced mathematical concepts. The ability to solve and graph linear inequalities is a cornerstone of mathematical proficiency. It equips us with the tools to analyze and interpret real-world scenarios involving constraints and ranges of values. From optimizing resource allocation to modeling physical phenomena, inequalities play a vital role in diverse fields. As we conclude this guide, it's important to recognize that the journey of mastering inequalities is an ongoing process. Continued practice and exploration will deepen your understanding and unlock new levels of mathematical fluency. The skills you've acquired here will serve as a stepping stone to more complex mathematical challenges, empowering you to tackle them with confidence and precision.

By mastering the techniques discussed, you can confidently solve and graph a wide range of linear inequalities.