Solving $-4x - 6 < -18$ And $7 - 12x > -65$ A Step-by-Step Guide

In the realm of mathematics, inequalities play a crucial role in defining ranges and constraints. Unlike equations that pinpoint specific values, inequalities deal with ranges of values, making them indispensable in various fields like optimization, economics, and computer science. When we encounter multiple inequalities linked together, we enter the domain of compound inequalities. This article will delve into the process of solving a compound inequality, using the example of $-4x - 6 < -18$ and $7 - 12x > -65$. We'll break down each step, ensuring clarity and comprehension, and highlight the underlying principles that govern these operations. Understanding compound inequalities is fundamental for anyone seeking to grasp the broader landscape of mathematical problem-solving.

Understanding Inequalities

Before we dive into solving the compound inequality, let's solidify our understanding of individual inequalities. An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations, which have definite solutions, inequalities often have a range of solutions. For instance, $x > 3$ means that x can be any value greater than 3, but not 3 itself. When working with inequalities, certain operations require special attention. Multiplying or dividing both sides of an inequality by a negative number necessitates flipping the inequality sign. This is a crucial rule to remember, as it directly impacts the solution set. Consider the inequality $-2x < 6$. If we divide both sides by -2, we must flip the inequality sign, resulting in $x > -3$. This seemingly small detail is pivotal in accurately determining the solution range. Furthermore, understanding how to represent inequalities on a number line provides a visual aid for grasping the solution set. An open circle indicates that the endpoint is not included (for < and >), while a closed circle indicates that it is (for ≤ and ≥). This visual representation can be particularly helpful when dealing with compound inequalities, where multiple ranges of values need to be considered. Inequalities are not just abstract mathematical constructs; they have practical applications in various real-world scenarios, from determining budget constraints to optimizing resource allocation. A strong foundation in inequalities is therefore essential for both academic pursuits and practical problem-solving.

Deconstructing the Compound Inequality: $-4x - 6 < -18$ and $7 - 12x > -65$

Our compound inequality presents us with two inequalities joined by the word "and": $-4x - 6 < -18$ and $7 - 12x > -65$. The presence of "and" is significant; it dictates that the solution set must satisfy both inequalities simultaneously. This means we're looking for values of 'x' that make both inequalities true. To tackle this, we'll treat each inequality separately, simplifying them step by step until we isolate 'x' in each case. The first inequality, $-4x - 6 < -18$, requires us to first isolate the term with 'x'. We can achieve this by adding 6 to both sides of the inequality: $-4x - 6 + 6 < -18 + 6$, which simplifies to $-4x < -12$. Next, we need to get 'x' by itself. This involves dividing both sides by -4. Remember the crucial rule: dividing by a negative number flips the inequality sign. So, $-4x < -12$ becomes $x > 3$. This tells us that one part of our solution requires 'x' to be greater than 3. Now, let's turn our attention to the second inequality, $7 - 12x > -65$. We begin by subtracting 7 from both sides: $7 - 12x - 7 > -65 - 7$, which simplifies to $-12x > -72$. Again, we need to isolate 'x'. We divide both sides by -12, remembering to flip the inequality sign: $-12x > -72$ becomes $x < 6$. This indicates that the other part of our solution requires 'x' to be less than 6. We've now successfully deconstructed the compound inequality into two simpler inequalities: $x > 3$ and $x < 6$. The next step is to combine these solutions to find the overall solution set.

Solving $-4x - 6 < -18$: A Step-by-Step Approach

To effectively solve the inequality $-4x - 6 < -18$, a systematic, step-by-step approach is paramount. This ensures accuracy and clarity in understanding the solution. Our primary goal is to isolate the variable 'x' on one side of the inequality. The first step in achieving this is to eliminate the constant term (-6) from the left side. We accomplish this by performing the inverse operation, which is adding 6 to both sides of the inequality. This maintains the balance of the inequality and allows us to simplify the expression. Adding 6 to both sides, we get: $-4x - 6 + 6 < -18 + 6$. This simplifies to $-4x < -12$. We have now successfully isolated the term containing 'x'. The next step involves getting 'x' completely by itself. Currently, 'x' is being multiplied by -4. To undo this multiplication, we need to perform the inverse operation, which is division. We divide both sides of the inequality by -4. However, and this is a crucial point, when we multiply or divide an inequality by a negative number, we must reverse the direction of the inequality sign. This is a fundamental rule in inequality manipulation. Dividing both sides by -4, we get: rac{-4x}{-4} > rac{-12}{-4}. Note that the '<' sign has been flipped to '>'. This simplifies to $x > 3$. This result tells us that any value of 'x' greater than 3 will satisfy the original inequality. The solution $x > 3$ represents a range of values, not a single value, which is characteristic of inequalities. Understanding this range is key to grasping the concept of inequalities. To further solidify our understanding, we can visualize this solution on a number line. We would draw an open circle at 3 (since x is strictly greater than 3, not equal to) and shade the region to the right, indicating all values greater than 3. This visual representation provides a clear picture of the solution set. This step-by-step method ensures that we arrive at the correct solution while understanding the underlying principles of inequality manipulation.

Solving $7 - 12x > -65$: A Detailed Explanation

Solving the inequality $7 - 12x > -65$ requires a similar step-by-step approach to the previous one, with careful attention to the rules governing inequality manipulations. Our objective remains the same: to isolate the variable 'x' on one side of the inequality. The first step involves addressing the constant term (7) on the left side. To eliminate it, we perform the inverse operation, which is subtracting 7 from both sides of the inequality. This maintains the balance and allows us to simplify the expression. Subtracting 7 from both sides, we get: $7 - 12x - 7 > -65 - 7$. This simplifies to $-12x > -72$. We have now isolated the term containing 'x'. The next step is to isolate 'x' completely. Currently, 'x' is being multiplied by -12. To undo this multiplication, we need to divide both sides of the inequality by -12. However, as we learned earlier, dividing an inequality by a negative number requires a crucial step: we must reverse the direction of the inequality sign. This is a fundamental rule that ensures the solution remains accurate. Dividing both sides by -12, we get: rac{-12x}{-12} < rac{-72}{-12}. Notice that the '>' sign has been flipped to '<'. This simplifies to $x < 6$. This result tells us that any value of 'x' less than 6 will satisfy the original inequality. This solution, $x < 6$, represents a range of values, consistent with the nature of inequalities. To visualize this solution, we can use a number line. We would draw an open circle at 6 (since x is strictly less than 6, not equal to) and shade the region to the left, indicating all values less than 6. This visual representation provides a clear understanding of the solution set. By carefully following these steps and remembering the rule about flipping the inequality sign when dividing by a negative number, we can confidently solve inequalities of this type.

Combining the Solutions: Finding the Intersection

Having solved each inequality separately, we now have two individual solutions: $x > 3$ and $x < 6$. The crucial connecting word in our original compound inequality was "and". This "and" signifies that the solution to the compound inequality is the intersection of the two individual solution sets. In other words, we need to find the values of 'x' that satisfy both $x > 3$ and $x < 6$ simultaneously. Visualizing these solutions on a number line is immensely helpful in determining the intersection. For $x > 3$, we have an open circle at 3 and shading to the right, indicating all values greater than 3. For $x < 6$, we have an open circle at 6 and shading to the left, indicating all values less than 6. The intersection is the region where the two shaded areas overlap. Examining the number line, we see that the overlap occurs between 3 and 6. This means the solution to the compound inequality is all values of 'x' that are greater than 3 and less than 6. We can express this solution in several ways. One way is to use inequality notation: $3 < x < 6$. This notation succinctly captures the fact that 'x' is bounded by 3 and 6. Another way to express the solution is using interval notation: $(3, 6)$. The parentheses indicate that the endpoints, 3 and 6, are not included in the solution set. This is because our original inequalities were strict inequalities ( > and < ), not inclusive ones ( ≥ and ≤ ). The interval notation provides a compact and widely understood way to represent solution sets. Understanding the concept of intersection and how it applies to compound inequalities is fundamental. It allows us to combine individual solutions into a comprehensive solution that satisfies all conditions.

Expressing the Solution: Inequality and Interval Notation

Once we've determined the solution set for a compound inequality, effectively communicating that solution is crucial. There are two primary methods for expressing solution sets: inequality notation and interval notation. Each has its advantages and is commonly used in mathematical contexts. As we established in the previous section, the solution to our compound inequality, $-4x - 6 < -18$ and $7 - 12x > -65$, is all values of 'x' that are greater than 3 and less than 6. Inequality notation provides a direct and intuitive way to represent this. We write the solution as $3 < x < 6$. This notation clearly states that 'x' is greater than 3 and simultaneously less than 6. It's a concise and easily understood representation. However, inequality notation can become cumbersome when dealing with more complex solution sets, such as those involving unions or unbounded intervals. This is where interval notation shines. Interval notation uses parentheses and brackets to indicate whether endpoints are included or excluded from the solution set. Parentheses, ( ), indicate that the endpoint is not included (corresponding to < and >), while brackets, [ ], indicate that the endpoint is included (corresponding to ≤ and ≥). For our solution, $3 < x < 6$, we use parentheses because 3 and 6 are not included. The interval notation for this solution is $(3, 6)$. The numbers 3 and 6 represent the lower and upper bounds of the interval, respectively. The use of parentheses signifies that the interval is open, meaning it does not include its endpoints. Interval notation is particularly useful for representing unbounded intervals. For example, the solution $x > 3$ would be written as $(3, ∞)$, where ∞ represents infinity. The parenthesis next to infinity always indicates that infinity is not included, as it's not a specific number but rather a concept. Similarly, $x ≤ 6$ would be written as $(-∞, 6]$. The bracket next to 6 indicates that 6 is included in the solution set. Mastering both inequality notation and interval notation is essential for clear and concise mathematical communication. Understanding when to use each notation enhances our ability to both express and interpret solutions effectively.

Visualizing the Solution on a Number Line

A powerful tool for understanding and communicating solutions to inequalities is the number line. Representing the solution set on a number line provides a visual aid that can clarify the range of values that satisfy the inequality. For our compound inequality, $-4x - 6 < -18$ and $7 - 12x > -65$, we found the solution to be $3 < x < 6$, which in interval notation is $(3, 6)$. To visualize this on a number line, we first draw a horizontal line representing all real numbers. We then mark the points 3 and 6 on the number line. Since our solution includes values strictly between 3 and 6 (i.e., not including 3 and 6 themselves), we use open circles at these points. An open circle indicates that the endpoint is not part of the solution set. If the solution had included 3 and 6 (i.e., $3 ≤ x ≤ 6$), we would use closed circles (or filled-in circles) to indicate that these points are included. Next, we shade the region between 3 and 6. This shaded region represents all the values of 'x' that satisfy the inequality $3 < x < 6$. Any point within this shaded region, when substituted for 'x' in the original compound inequality, will make both inequalities true. The number line provides a clear visual representation of the solution set. It allows us to quickly grasp the range of values that are included in the solution and those that are excluded. For instance, we can easily see that 4 and 5 are part of the solution, while 2 and 7 are not. Visualizing solutions on a number line is particularly helpful when dealing with compound inequalities involving "or". In such cases, the solution may consist of two or more disjoint intervals, which are readily represented on a number line. The number line is a valuable tool not just for representing solutions but also for checking them. We can pick a value within the shaded region and substitute it into the original inequality to verify that it satisfies the conditions. Similarly, we can pick a value outside the shaded region and confirm that it does not satisfy the inequality. This visual verification can help catch errors and build confidence in our solution. In conclusion, visualizing solutions on a number line is a crucial skill for anyone working with inequalities. It provides a clear and intuitive way to understand and communicate solution sets.

Conclusion: Mastering Compound Inequalities

In this comprehensive guide, we've dissected the process of solving a compound inequality, using the example of $-4x - 6 < -18$ and $7 - 12x > -65$. We've walked through each step, from isolating the variable in individual inequalities to combining the solutions and expressing them in both inequality and interval notation. We've also emphasized the importance of visualizing solutions on a number line. Mastering compound inequalities is a valuable skill in mathematics, with applications extending to various fields. Understanding how to manipulate inequalities, remember the critical rule of flipping the sign when multiplying or dividing by a negative number, and interpreting the logical connective "and" are all essential components of this skill. The step-by-step approach we've outlined provides a framework for tackling similar problems with confidence. By breaking down complex inequalities into simpler steps, we can avoid errors and arrive at accurate solutions. Visual aids like the number line further enhance our understanding and allow us to communicate solutions effectively. Beyond the specific example we've addressed, the principles and techniques discussed here are broadly applicable to a wide range of inequality problems. Whether you're dealing with linear inequalities, quadratic inequalities, or systems of inequalities, a solid foundation in these fundamental concepts will serve you well. Practice is key to mastering any mathematical skill, and working through various examples of compound inequalities will solidify your understanding. As you gain experience, you'll develop an intuition for the behavior of inequalities and become more adept at solving them. Ultimately, the ability to solve compound inequalities is not just about finding the right answer; it's about developing a logical and systematic approach to problem-solving, a skill that is valuable in all aspects of life.