Graphing Inequalities: A Step-by-Step Solution

avigating the world of inequalities can sometimes feel like traversing a maze. But fear not! This comprehensive guide will walk you through the process of graphing the solution to the inequality $\frac{4}{9} x - 10 > \frac{x}{3} - 12$ step by step. We'll break down the equation, simplify it, and finally, represent the solution on a graph. So, let's put on our mathematical hats and dive in!

Understanding Inequalities

Before we jump into the specific problem, let's take a moment to understand what inequalities are. Unlike equations that show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols used to represent these relationships are: > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). Grasping this fundamental concept is crucial for effectively graphing inequality solutions.

Inequalities are mathematical statements that compare two expressions using inequality symbols. When dealing with inequalities, we're not just looking for a single solution, but rather a range of values that satisfy the given condition. This range of values is what we represent graphically. To successfully graph the solution of an inequality, you need to isolate the variable on one side and then represent the solution set on a number line. This process involves understanding the properties of inequalities and how they differ from equations. For example, multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. This is a critical rule to remember as it directly impacts the solution set you'll be graphing. Inequalities play a vital role in various fields, from economics to engineering, making their graphical representation a valuable skill to master. So, let's dive deeper into the process and understand how to visualize these solutions effectively.

Step 1: Simplify the Inequality

The first step in solving any inequality is to simplify it. This involves combining like terms and isolating the variable on one side of the inequality. Our inequality is: $\frac{4}{9} x - 10 > \frac{x}{3} - 12$. To get rid of the fractions, we can multiply both sides of the inequality by the least common multiple (LCM) of 9 and 3, which is 9. This eliminates the fractions and makes the inequality easier to work with. Remember, whatever operation you perform on one side of the inequality, you must also perform on the other side to maintain the balance. The key here is to simplify the inequality into a form that's easier to solve, and eliminating fractions is a significant step in that direction. This ensures that we are working with whole numbers, reducing the chances of making errors in the subsequent steps. By meticulously simplifying the inequality, we set the stage for accurately determining the range of values that satisfy the original condition.

Multiplying by the LCM

Multiplying both sides by 9, we get: $9 * (\frac{4}{9} x - 10) > 9 * (\frac{x}{3} - 12)$. Distributing the 9 on both sides gives us: $4x - 90 > 3x - 108$. Now, we have an inequality without fractions, which is much easier to handle. This step highlights the importance of identifying the LCM, which is a fundamental skill in working with fractions and inequalities. It's essential to ensure that the multiplication is carried out correctly on both sides of the inequality to preserve its validity. This process not only simplifies the expression but also sets the foundation for isolating the variable and ultimately determining the solution set.

Step 2: Isolate the Variable

Now that we've eliminated the fractions, let's isolate the variable x. To do this, we need to get all the terms with x on one side of the inequality and all the constant terms on the other side. In our inequality, $4x - 90 > 3x - 108$, we can subtract 3x from both sides to get the x terms on the left side. Then, we can add 90 to both sides to get the constant terms on the right side. Remember, the goal is to manipulate the inequality while maintaining its truth. This step is a crucial part of the process, as it brings us closer to defining the range of values that x can take to satisfy the inequality.

Moving Terms

Subtracting 3x from both sides, we have: $4x - 3x - 90 > 3x - 3x - 108$, which simplifies to $x - 90 > -108$. Next, adding 90 to both sides, we get: $x - 90 + 90 > -108 + 90$, which simplifies to $x > -18$. We've successfully isolated x! This is a significant milestone because we now have a clear understanding of the condition that x must satisfy. The inequality $x > -18$ tells us that the solution set includes all values of x that are greater than -18. This is a critical piece of information that will guide us in the next step, which is graphing the solution.

Step 3: Graph the Solution

With the variable isolated, we can now graph the solution on a number line. The inequality $x > -18$ tells us that the solution includes all numbers greater than -18. To represent this on a number line, we'll draw a number line and mark -18 on it. Since the inequality is strictly “greater than” (not greater than or equal to), we'll use an open circle at -18 to indicate that -18 itself is not included in the solution. Then, we'll draw an arrow extending to the right from -18 to represent all numbers greater than -18. This graphical representation provides a visual understanding of the solution set, making it easy to see which values satisfy the inequality. Graphing the solution is a powerful tool for visualizing inequalities and communicating their solutions effectively.

Representing on the Number Line

On the number line, place an open circle at -18. This signifies that -18 is not included in the solution set. Then, draw an arrow extending from -18 towards the right, indicating all the numbers greater than -18. This arrow represents the infinite number of solutions that satisfy the inequality $x > -18$. The graph serves as a clear and concise visual representation of the solution, making it easier to understand the range of values that make the inequality true. This visual aid is particularly helpful in more complex scenarios where multiple inequalities or compound inequalities are involved. The ability to translate an algebraic inequality into a graphical representation is a fundamental skill in mathematics.

Step 4: Verifying the Solution

To ensure that our solution is correct, it's always a good idea to verify it. We can do this by choosing a value from the solution set (a number greater than -18) and plugging it back into the original inequality. If the inequality holds true, then our solution is likely correct. Similarly, we can choose a value outside the solution set (a number less than or equal to -18) and plug it into the original inequality. If the inequality does not hold true, this further confirms the correctness of our solution. This verification step is a crucial part of the problem-solving process, as it helps to catch any potential errors and ensures that the final answer is accurate.

Testing Values

Let's choose 0, which is greater than -18. Plugging 0 into the original inequality, we get: $\frac{4}{9} (0) - 10 > \frac{0}{3} - 12$, which simplifies to $-10 > -12$. This is true, so our solution is likely correct. Now, let's choose -19, which is less than -18. Plugging -19 into the original inequality, we get: $\frac{4}{9} (-19) - 10 > \frac{-19}{3} - 12$, which simplifies to approximately -18.44 > -18.33. This is false, further confirming that our solution $x > -18$ is correct. By systematically testing values within and outside the solution set, we gain confidence in the accuracy of our result. This method of verification is a valuable tool for anyone working with inequalities, as it provides a means of checking the solution's validity.

Conclusion

Graphing the solution of the inequality $\frac{4}{9} x - 10 > \frac{x}{3} - 12$ involves simplifying the inequality, isolating the variable, representing the solution on a number line, and verifying the solution. By following these steps, you can confidently solve and graph inequalities. Remember, practice makes perfect, so keep working on different problems to strengthen your skills! Understanding how to graph inequalities is not only essential for math class but also has real-world applications in various fields, from finance to engineering. The ability to visualize inequalities and their solutions provides a valuable tool for problem-solving and decision-making. So, keep practicing, and you'll become a master of inequalities!

For further exploration of inequalities and their applications, you might find helpful resources on websites like Khan Academy.