Solving Inequalities A Step-by-Step Guide To Solve -3(3h-1) ≤ -7h - 5

Introduction

In this article, we will walk through the process of solving the inequality $-3(3h - 1) \leq -7h - 5$ for $h$. Inequalities are mathematical statements that compare two expressions using inequality symbols such as less than ($\lt$), greater than ($>$), less than or equal to ($\leq$), and greater than or equal to ($\geq$). Solving an inequality involves finding the range of values for a variable that makes the inequality true. This particular inequality involves a linear expression, and we will use algebraic techniques to isolate $h$ and find the solution set. Understanding how to solve inequalities is crucial in various fields of mathematics and real-world applications, including optimization problems, economics, and physics. This step-by-step guide will help you grasp the fundamental principles and techniques for solving similar problems. This introduction sets the stage for a detailed exploration of the solution process, ensuring that you understand not only the steps but also the underlying mathematical concepts. Mastering the solution of inequalities is a fundamental skill in algebra, with applications spanning across various mathematical domains and practical scenarios.

The ability to solve inequalities is a cornerstone of mathematical proficiency, opening doors to more advanced topics and real-world applications. Before diving into the specifics of this problem, it's worth noting the broader importance of inequalities in mathematics and beyond. Inequalities are used extensively in calculus, where they play a vital role in defining limits, continuity, and convergence. In optimization problems, inequalities help define constraints, ensuring that solutions meet specific criteria or limitations. Economists use inequalities to model scenarios involving resource allocation, supply and demand, and economic equilibrium. In physics, inequalities can be used to describe physical constraints, such as the maximum velocity of an object or the range of acceptable temperatures in an experiment. Therefore, understanding and mastering the techniques for solving inequalities is not just an academic exercise but a valuable skill that will serve you well in a wide array of contexts. By the end of this article, you will have a clear understanding of how to approach and solve linear inequalities, equipping you with the confidence and skills to tackle more complex problems in the future.

Understanding the Basics of Inequalities

Before we jump into solving the inequality, let's briefly discuss the basics of inequalities. An inequality is a mathematical statement that compares two expressions using symbols such as $\leq$ (less than or equal to), $\geq$ (greater than or equal to), $<$ (less than), and $>$ (greater than). Unlike equations, which have a specific solution or set of solutions, inequalities often have a range of solutions. For example, the inequality $x > 3$ means that any value of $x$ greater than 3 is a solution. When solving inequalities, we aim to isolate the variable on one side of the inequality to determine the range of values that satisfy the condition. This process often involves applying algebraic operations such as addition, subtraction, multiplication, and division. However, there's a crucial rule to remember: when multiplying or dividing both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. This rule is essential for maintaining the correctness of the solution. For instance, if we have $-2x < 6$, dividing both sides by -2 requires us to flip the inequality sign, resulting in $x > -3$. Failing to do so would lead to an incorrect solution. Understanding this rule and the basic properties of inequalities is fundamental for solving more complex problems. This foundational knowledge allows us to approach inequalities with confidence, ensuring we apply the correct techniques to find accurate solutions. Inequalities are not just abstract mathematical concepts; they have real-world applications in various fields, including economics, physics, and computer science, making their understanding even more critical.

Step-by-Step Solution

Now, let's solve the inequality step by step:

Step 1: Distribute the -3

The first step in solving the inequality $-3(3h - 1) \leq -7h - 5$ is to distribute the $-3$ across the terms inside the parentheses. This involves multiplying $-3$ by both $3h$ and $-1$. Performing this operation, we get:

$$-3 * (3h) + (-3) * (-1) \leq -7h - 5$$

Simplifying this, we obtain:

$$-9h + 3 \leq -7h - 5$$

Distributing the constant across the parentheses is a fundamental algebraic technique used to eliminate parentheses and simplify expressions. This step is crucial because it allows us to combine like terms and isolate the variable $h$. By correctly applying the distributive property, we transform the original inequality into a more manageable form. Without this step, it would be difficult to proceed with solving for $h$. The distributive property, which states that $a(b + c) = ab + ac$, is a cornerstone of algebraic manipulation. Mastering this property ensures that you can accurately simplify expressions and equations, making subsequent steps in the solution process more straightforward. In this case, multiplying $-3$ by both $3h$ and $-1$ is essential for unraveling the inequality and preparing it for further simplification. This step sets the foundation for the subsequent steps, where we will isolate the variable $h$ and determine the solution set.

Step 2: Add $9h$ to Both Sides

To isolate the terms containing $h$, we need to eliminate $-9h$ from the left side of the inequality. We can do this by adding $9h$ to both sides of the inequality. This ensures that the inequality remains balanced. Adding $9h$ to both sides gives us:

$$-9h + 3 + 9h \leq -7h - 5 + 9h$$

Simplifying this, we get:

$$3 \leq 2h - 5$$

Adding the same term to both sides of an inequality is a fundamental property of inequalities. This property allows us to manipulate the inequality while preserving the relationship between the two sides. In this case, adding $9h$ to both sides helps us consolidate the $h$ terms on one side of the inequality, making it easier to isolate the variable. This step is crucial for progressing towards the solution. Maintaining balance in an inequality is paramount; any operation performed on one side must also be performed on the other. This ensures that the inequality remains valid and the solution set remains unchanged. By carefully adding $9h$ to both sides, we eliminate the $-9h$ term on the left side, simplifying the inequality and bringing us closer to isolating $h$. This step demonstrates a key algebraic technique for solving inequalities and equations alike, highlighting the importance of performing operations on both sides to maintain equivalence.

Step 3: Add 5 to Both Sides

Next, we want to isolate the term with $h$ on the right side. To do this, we add 5 to both sides of the inequality:

$$3 + 5 \leq 2h - 5 + 5$$

Simplifying, we have:

$$8 \leq 2h$$

Adding a constant to both sides of an inequality is a valid operation that helps isolate the variable term. In this specific step, we add 5 to both sides of the inequality to eliminate the constant term on the right side, bringing us closer to isolating $h$. This process is crucial for solving inequalities because it simplifies the expression and makes it easier to determine the range of values for $h$ that satisfy the inequality. By adding 5 to both sides, we maintain the balance of the inequality, ensuring that the solution set remains unchanged. This step highlights the importance of performing inverse operations to isolate variables, a fundamental technique in algebra. The goal is to gradually strip away the terms surrounding the variable until the variable is by itself on one side of the inequality. Adding 5 is a key step in this process, as it removes the constant term, making the next step—dividing by the coefficient of $h$—more straightforward.

Step 4: Divide Both Sides by 2

To finally isolate $h$, we divide both sides of the inequality by 2:

$$\frac{8}{2} \leq \frac{2h}{2}$$

This simplifies to:

$$4 \leq h$$

Dividing both sides of an inequality by a positive number is a fundamental algebraic operation that helps isolate the variable and determine the solution set. In this step, we divide both sides of the inequality $8 \leq 2h$ by 2, which is the coefficient of $h$. This action effectively isolates $h$ on the right side of the inequality, bringing us to the final solution. It's important to note that when dividing or multiplying both sides of an inequality by a negative number, we would need to reverse the inequality sign. However, since we are dividing by a positive number (2), the inequality sign remains the same. This step is a crucial part of the solution process because it provides the direct relationship between $h$ and a constant value, allowing us to understand the range of values that satisfy the inequality. Dividing by the coefficient of the variable is a common technique in solving both equations and inequalities, and mastering this step is essential for success in algebra. The result, $4 \leq h$, tells us that $h$ must be greater than or equal to 4 for the original inequality to hold true.

Final Answer

Thus, the solution to the inequality $-3(3h - 1) \leq -7h - 5$ is:

$$h \geq 4$$

This means that any value of $h$ that is greater than or equal to 4 will satisfy the original inequality. We can also represent this solution in interval notation as $[4, \infty)$. This notation indicates that the solution includes all real numbers from 4 (inclusive) to infinity. The final answer, $h \geq 4$, represents the culmination of the step-by-step algebraic manipulations we performed to isolate $h$. This solution is a concise way of stating the range of values that satisfy the original inequality. Understanding the solution in this form is crucial for interpreting the results and applying them in various contexts. Additionally, expressing the solution in interval notation, $[4, \infty)$, provides a clear and standardized way to represent the solution set graphically and mathematically. The square bracket on the left indicates that 4 is included in the solution, while the parenthesis on the right signifies that infinity is not a specific endpoint but rather an unbounded continuation. This comprehensive understanding of the solution, both in inequality and interval notation, ensures that the results are clear, accurate, and applicable to further mathematical analysis or real-world problem-solving. By arriving at this final answer, we have successfully navigated the algebraic steps necessary to solve the inequality and have a clear understanding of the solution set.

Conclusion

In conclusion, we have successfully solved the inequality $-3(3h - 1) \leq -7h - 5$ for $h$, and found the solution to be $h \geq 4$. This process involved distributing, combining like terms, and isolating the variable. Understanding how to solve inequalities is a fundamental skill in algebra and is essential for various mathematical applications. The conclusion of this article serves as a recap of the entire solution process and highlights the key steps involved in solving the inequality. We started by distributing the constant, then combined like terms by adding and subtracting values from both sides, and finally isolated the variable by dividing. Each step was crucial in simplifying the inequality and arriving at the correct solution. The final solution, $h \geq 4$, provides a clear and concise answer to the problem, indicating the range of values for $h$ that satisfy the inequality. This conclusion also underscores the importance of mastering inequality-solving techniques as a fundamental skill in algebra. Inequalities are not just abstract mathematical concepts; they have practical applications in various fields, including economics, physics, and engineering. Being able to solve inequalities equips you with the tools to analyze and solve real-world problems that involve constraints and limitations. Therefore, understanding and practicing these techniques will enhance your mathematical proficiency and your ability to apply mathematical concepts in diverse contexts. This article has provided a detailed walkthrough of the solution process, and with continued practice, you can confidently tackle similar problems in the future.

Importance of Mastering Inequalities

Mastering inequalities is crucial because they appear in various mathematical contexts and real-world applications. From calculus to optimization problems, inequalities are fundamental tools for problem-solving. They help define constraints, establish boundaries, and determine feasible regions in mathematical models. In real-world scenarios, inequalities are used in economics to model supply and demand, in physics to describe physical limitations, and in computer science to analyze algorithm efficiency. Therefore, a strong understanding of inequalities is not just an academic asset but a practical skill that enhances problem-solving abilities across diverse fields.

Further Practice

To reinforce your understanding, try solving similar inequalities. Practice is key to mastering algebraic techniques. You can find numerous examples in textbooks, online resources, and practice worksheets. Working through a variety of problems will help you become more comfortable with the steps involved and improve your problem-solving speed and accuracy. Additionally, consider exploring inequalities in different contexts, such as compound inequalities and absolute value inequalities, to broaden your knowledge and skills.