Solving -3/4x + 2 <= -7 A Step-by-Step Guide

In the realm of mathematics, inequalities play a pivotal role, extending the concept of equality to encompass a range of values. Understanding how to solve inequalities is fundamental for various applications, from optimizing solutions in linear programming to defining feasible regions in calculus. This article delves into a step-by-step guide on solving the inequality $-\frac{3}{4}x + 2 \leq -7$, providing a comprehensive understanding of the underlying principles and techniques involved. This detailed explanation not only aids in grasping the solution but also enhances the problem-solving skills necessary for tackling more complex mathematical challenges. We aim to break down each step, ensuring clarity and reinforcing the fundamental concepts necessary for manipulating inequalities effectively. Whether you're a student grappling with algebra or someone seeking to refresh their mathematical skills, this article will serve as an invaluable resource.

The core concept of solving inequalities revolves around isolating the variable, much like solving equations. However, a crucial distinction arises when multiplying or dividing by a negative number, which necessitates flipping the inequality sign. This article will meticulously explore this rule and its application, ensuring a thorough understanding of its significance. Furthermore, we'll emphasize the importance of verifying the solution set, which ensures that the derived values satisfy the original inequality. By the end of this guide, you will not only be able to solve this particular inequality but also possess the confidence and knowledge to approach a wide array of similar problems. The journey through this solution will illuminate the logical progression and the necessary algebraic manipulations, reinforcing your understanding of mathematical principles.

The process of solving inequalities is not just about arriving at an answer; it's about understanding the logical flow and the implications of each step. This article aims to foster that understanding by providing a detailed explanation of each manipulation. We'll start by simplifying the inequality, moving constants to one side, and then isolating the variable term. Following this, we'll address the fractional coefficient and the crucial step of handling the negative sign. Each of these steps will be elaborated upon with clear reasoning and examples, ensuring that the reader grasps not only the "how" but also the "why" behind each action. This approach will empower you to apply these techniques to diverse problems and build a solid foundation in algebraic manipulation. Let's embark on this mathematical journey together and unravel the solution to this inequality, enhancing your mathematical acumen along the way.

Step 1: Isolate the Variable Term

To solve the inequality $-\frac{3}{4}x + 2 \leq -7$, our initial goal is to isolate the term containing the variable, which in this case is $-\frac{3}{4}x$. This is achieved by performing operations that maintain the inequality's balance, similar to how equations are handled. The first step involves eliminating the constant term, which is +2, from the left side of the inequality. To do this, we subtract 2 from both sides of the inequality. This operation ensures that the inequality remains valid because subtracting the same value from both sides does not alter the relationship between them. This is a fundamental principle in manipulating inequalities and equations alike. Understanding this principle is crucial for correctly applying algebraic techniques and arriving at accurate solutions. The subtraction effectively cancels out the +2 on the left side, leaving us with the variable term and a new constant on the right side. This step sets the stage for further isolating the variable and ultimately determining the solution set for the inequality.

The mathematical rationale behind subtracting 2 from both sides stems from the properties of inequalities. Specifically, the addition/subtraction property of inequality states that if a < b, then a + c < b + c and a - c < b - c for any real number c. This property allows us to manipulate inequalities by adding or subtracting the same value from both sides without changing the direction of the inequality. In our case, subtracting 2 from both sides allows us to isolate the term containing 'x' on one side, bringing us closer to the solution. The resulting inequality, $-\frac{3}{4}x \leq -9$, is a simplified form of the original inequality and is easier to work with. It is crucial to remember that each step in solving an inequality must be mathematically sound and based on established principles. This ensures that the final solution is accurate and that the process can be applied to other inequalities as well.

After subtracting 2 from both sides, the inequality transforms to $-\frac{3}{4}x \leq -9$. This simplified inequality is now in a form where we can directly address the coefficient of x. The next step involves dealing with the fraction and the negative sign attached to the 'x' term. This is a critical step as it requires careful attention to the rules governing inequality manipulation. The process of isolating the variable 'x' involves understanding the multiplicative inverse and its effect on the inequality sign. The overall strategy is to transform the inequality into a form where 'x' stands alone on one side, clearly indicating the range of values that satisfy the original inequality. This step-by-step approach ensures clarity and reduces the chances of errors in the solution process. The goal is to make the process transparent and easy to follow, ultimately enhancing the understanding of inequality manipulation.

Step 2: Multiply by the Reciprocal and Flip the Inequality Sign

The next critical step in solving the inequality $-\frac{3}{4}x \leq -9$ is to isolate the variable x. To achieve this, we need to eliminate the coefficient $-\frac{3}{4}$. The most efficient way to do this is by multiplying both sides of the inequality by the reciprocal of $-\frac{3}{4}$, which is $-\frac{4}{3}$. This multiplication will effectively cancel out the coefficient on the left side, leaving x isolated. However, a crucial rule in dealing with inequalities comes into play here: when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is a fundamental principle that ensures the inequality remains mathematically valid. The reason behind this rule is rooted in the nature of negative numbers and their effect on ordering. Multiplying by a negative number essentially flips the number line, causing values to switch their positions relative to each other. Therefore, the inequality sign must be flipped to maintain the correct relationship.

The mathematical principle that dictates the flipping of the inequality sign when multiplying or dividing by a negative number is essential to understand. Consider a simple example: 2 < 4. If we multiply both sides by -1, we get -2 and -4. Clearly, -2 is greater than -4, so the inequality sign must be flipped to maintain the correct relationship: -2 > -4. This same principle applies to more complex inequalities. When we multiply both sides of $-\frac{3}{4}x \leq -9$ by $-\frac{4}{3}$, we are multiplying by a negative number, hence the need to flip the "less than or equal to" sign to "greater than or equal to." Failing to do so would result in an incorrect solution set. The act of flipping the inequality sign is not just a mechanical step; it is a direct consequence of the properties of negative numbers and their interaction with inequalities. Understanding this principle solidifies the foundation for solving inequalities accurately and confidently.

By multiplying both sides of $-\frac{3}{4}x \leq -9$ by $-\frac{4}{3}$ and flipping the inequality sign, we obtain the expression $x \geq (-9) imes (-\frac{4}{3})$. This step sets up the final calculation to determine the range of values for x that satisfy the original inequality. The multiplication on the right side involves two negative numbers, which will result in a positive number. This is another key concept in dealing with negative numbers and arithmetic operations. The next step will be to simplify this expression to find the numerical value that defines the lower bound for x. This process highlights the importance of carefully following the order of operations and paying attention to the signs of the numbers involved. The ultimate goal is to arrive at a clear and concise solution that accurately represents the solution set for the inequality.

Step 3: Simplify the Expression

Following the multiplication by the reciprocal and the crucial flipping of the inequality sign, we arrive at the expression $x \geq (-9) imes (-\frac{4}{3})$. The next step is to simplify this expression to determine the final solution for x. The right side of the inequality involves the product of two negative numbers, which, according to the rules of arithmetic, results in a positive number. Specifically, we are multiplying -9 by -$\frac{4}{3}$. To perform this multiplication, it can be helpful to express -9 as a fraction, which is -$\frac{9}{1}$. Then, we can multiply the numerators and the denominators separately. This process involves multiplying -9 by -4, which equals 36, and multiplying 1 by 3, which equals 3. The result is the fraction $\frac{36}{3}$. This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. This simplification process is a fundamental skill in arithmetic and algebra and is crucial for expressing solutions in their simplest form. The ability to accurately simplify expressions ensures clarity and facilitates further mathematical operations.

Simplifying the fraction $\frac{36}{3}$ involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In this case, the GCD of 36 and 3 is 3. Dividing 36 by 3 gives us 12, and dividing 3 by 3 gives us 1. Therefore, the simplified fraction is $\frac{12}{1}$, which is equivalent to the whole number 12. This simplification leads us to the final form of the inequality: $x \geq 12$. This result indicates that the solution set for the original inequality includes all real numbers that are greater than or equal to 12. This is a concise and clear representation of the solution, and it is essential for understanding the range of values that satisfy the given inequality. The process of simplification not only makes the solution easier to interpret but also reduces the chances of errors in subsequent calculations or applications of the result. The simplified form $x \geq 12$ is the key to understanding the solutions to the inequality.

Thus, the simplified expression reveals the solution to the inequality: $x \geq 12$. This means that x can be any number that is greater than or equal to 12. This solution can be visualized on a number line, where a closed circle is placed at 12, and an arrow extends to the right, indicating that all values to the right of 12, including 12 itself, are part of the solution set. This visual representation provides a clear understanding of the range of values that satisfy the inequality. The solution $x \geq 12$ represents an infinite set of values, all of which, when substituted into the original inequality, will result in a true statement. This comprehensive understanding of the solution set is crucial for applying this result in various mathematical and real-world contexts. The solution $x \geq 12$ not only answers the specific problem but also provides valuable insights into the behavior of inequalities and their applications.

Solution

Therefore, the solution to the inequality $-\frac{3}{4}x + 2 \leq -7$ is $x \geq 12$. This result signifies that any value of x that is greater than or equal to 12 will satisfy the original inequality. To reiterate, we arrived at this solution by systematically isolating the variable x through a series of algebraic manipulations. First, we subtracted 2 from both sides of the inequality to isolate the term containing x. Then, we multiplied both sides by the reciprocal of the coefficient of x, which was $-\frac{4}{3}$. A critical step in this process was flipping the inequality sign because we multiplied by a negative number. Finally, we simplified the resulting expression to obtain the solution $x \geq 12$. This solution is a concise and unambiguous representation of the range of values that make the inequality true. It is crucial to understand that the solution is not just a single number but a set of numbers, all of which meet the condition specified by the inequality.

To validate our solution, we can test a value within the solution set and a value outside the solution set in the original inequality. For instance, let's test x = 12, which is the boundary value of our solution set. Substituting x = 12 into the original inequality, we get $-\frac{3}{4}(12) + 2 \leq -7$. Simplifying this expression, we have -9 + 2 \leq -7, which further simplifies to -7 \leq -7. This is a true statement, confirming that 12 is indeed part of the solution set. Now, let's test a value greater than 12, say x = 16. Substituting x = 16, we get $-\frac{3}{4}(16) + 2 \leq -7$, which simplifies to -12 + 2 \leq -7, and further to -10 \leq -7. This is also a true statement, indicating that values greater than 12 are also part of the solution set. To test a value outside the solution set, let's try x = 8. Substituting x = 8, we get $-\frac{3}{4}(8) + 2 \leq -7$, which simplifies to -6 + 2 \leq -7, and further to -4 \leq -7. This is a false statement, confirming that values less than 12 are not part of the solution set. These tests provide strong evidence that our solution $x \geq 12$ is accurate and encompasses all values that satisfy the original inequality.

The implications of the solution $x \geq 12$ extend beyond the immediate problem. Understanding how to solve inequalities is a fundamental skill in mathematics and has wide-ranging applications in various fields. Inequalities are used to model real-world situations where constraints and limitations exist, such as in optimization problems, resource allocation, and decision-making processes. For example, in business, inequalities can be used to determine the range of production levels that will result in a profit. In engineering, they can be used to ensure that designs meet certain safety standards. In economics, inequalities are used to model supply and demand relationships. The ability to solve inequalities accurately and efficiently is therefore a valuable asset in many disciplines. Moreover, the process of solving inequalities reinforces critical thinking and problem-solving skills, which are essential for success in any field. The solution $x \geq 12$ is not just an answer to a mathematical problem; it is a gateway to understanding and applying mathematical concepts in a broader context. This detailed exploration of the solution and its implications underscores the importance of mastering the fundamentals of algebra and their relevance to real-world applications.

In conclusion, solving the inequality $-\frac{3}{4}x + 2 \leq -7$ has provided a valuable exercise in understanding and applying the principles of algebraic manipulation. Through a step-by-step process, we isolated the variable x and determined that the solution set is $x \geq 12$. This solution represents all real numbers greater than or equal to 12, a range of values that satisfy the original inequality. The process involved subtracting a constant from both sides, multiplying by the reciprocal of a fraction (and remembering to flip the inequality sign), and simplifying the resulting expression. Each step was carefully explained to ensure clarity and reinforce the underlying mathematical concepts. The importance of flipping the inequality sign when multiplying or dividing by a negative number was emphasized, as this is a critical rule that distinguishes inequality manipulation from equation solving. The solution was also validated by testing values within and outside the solution set, providing further confidence in its accuracy.

This comprehensive exploration of solving the inequality serves as a foundation for tackling more complex mathematical problems. The skills and techniques learned in this exercise are applicable to a wide range of algebraic problems, including those involving more variables, systems of inequalities, and different types of inequalities (e.g., quadratic, absolute value). Furthermore, the ability to solve inequalities is essential for various applications in mathematics, science, engineering, economics, and other fields. Inequalities are used to model constraints, optimize solutions, and make informed decisions in real-world scenarios. The process of solving inequalities also reinforces critical thinking and problem-solving skills, which are valuable assets in any discipline. By mastering the fundamentals of inequality manipulation, individuals can confidently approach a diverse array of challenges and effectively apply mathematical concepts to solve them.

Ultimately, the solution $x \geq 12$ is not just an answer to a specific problem; it is a testament to the power of algebraic techniques and the importance of understanding fundamental mathematical principles. This article has aimed to provide not only the solution but also a deep understanding of the process and the underlying logic. By mastering the art of solving inequalities, individuals can unlock a new level of mathematical proficiency and apply these skills to a wide range of real-world applications. The journey through this solution has hopefully illuminated the beauty and power of mathematics, inspiring further exploration and learning in this fascinating field. The ability to solve inequalities is a valuable tool in the mathematical arsenal, empowering individuals to tackle challenges with confidence and precision.