Solving Inequalities A Step-by-Step Guide To $2(4+2x) \geq 5x+5$

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Understanding how to solve inequalities is fundamental for various applications, from optimizing resource allocation to modeling real-world constraints. This article delves into the process of solving the linear inequality $2(4+2x) \geq 5x+5$, providing a detailed, step-by-step explanation to help you master this essential skill. We will explore the underlying principles, demonstrate each step with clarity, and discuss the significance of the solution in the context of mathematical problem-solving. So, let's embark on this journey to unravel the intricacies of inequalities and equip ourselves with the tools to tackle them confidently.

Understanding Inequalities and Their Significance

Before we dive into the specific problem, let's first understand the concept of inequalities and why they are so important in mathematics. Unlike equations, which represent equality between two expressions, inequalities express a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. These relationships are represented by the symbols >, <, ${\geq}$, and ${\leq}$, respectively. Inequalities are used extensively in various mathematical fields, including calculus, linear programming, and real analysis. They allow us to model situations where exact solutions may not exist or are not practical, and provide a range of possible values that satisfy a given condition. For example, in economics, inequalities can be used to describe budget constraints or production capacities. In physics, they can represent the range of possible values for physical quantities such as velocity or energy. The ability to solve inequalities is therefore crucial for anyone seeking to apply mathematical principles to real-world problems.

Step 1: Distribute the Multiplication

The first step in solving the inequality $2(4+2x) \geq 5x+5$ is to simplify the expression by distributing the multiplication on the left side. This involves multiplying the number outside the parentheses, which is 2, by each term inside the parentheses, which are 4 and 2x. This step is based on the distributive property of multiplication over addition, which states that a(b + c) = ab + ac. Applying this property to our inequality, we get:

• 2 * 4 = 8
• 2 * 2x = 4x

Therefore, the left side of the inequality becomes 8 + 4x. Now, we can rewrite the inequality as:

8 + 4x \geq 5x + 5

This simplified form is much easier to work with and allows us to proceed with the next steps in solving the inequality. Distributing the multiplication is a crucial first step in many algebraic problems, as it helps to eliminate parentheses and combine like terms, making the equation or inequality more manageable.

Step 2: Combine Like Terms – Isolating the Variable

Now that we've distributed the multiplication, our inequality looks like this: $8 + 4x \geq 5x + 5$. The next step is to combine like terms, which means gathering all the terms containing the variable x on one side of the inequality and all the constant terms (numbers) on the other side. This process is crucial for isolating the variable and ultimately finding the solution. To do this, we can perform the same operations on both sides of the inequality, just as we would with an equation. Our goal is to get all the x terms on one side and the constant terms on the other. Let's start by subtracting 4x from both sides of the inequality. This will eliminate the x term on the left side:

8 + 4x - 4x \geq 5x + 5 - 4x

Simplifying this, we get:

8 \geq x + 5

Next, we need to isolate the x term further by removing the constant term (+5) from the right side. We can do this by subtracting 5 from both sides of the inequality:

8 - 5 \geq x + 5 - 5

Simplifying again, we get:

3 \geq x

This is a crucial step because we have now isolated the variable x on one side of the inequality. This simplified form makes it much easier to interpret the solution. Combining like terms is a fundamental technique in algebra and is essential for solving equations and inequalities efficiently.

Step 3: Interpreting the Solution

After combining like terms, we've arrived at the inequality $3 \geq x$. This inequality states that 3 is greater than or equal to x. However, it's often more intuitive to read the inequality with the variable on the left side. We can rewrite the inequality as:

x \leq 3

This is read as "x is less than or equal to 3." This form clearly shows us the range of values that x can take to satisfy the original inequality. The solution includes all real numbers that are 3 or less. This means that any value of x that is 3, or any number smaller than 3, will make the original inequality true. For example, if we substitute x = 3 into the original inequality, we get:

• 2(4 + 2 * 3) \geq 5 * 3 + 5
• 2(4 + 6) \geq 15 + 5
• 2(10) \geq 20
• 20 \geq 20

This is true, so x = 3 is part of the solution. Similarly, if we substitute x = 0, we get:

• 2(4 + 2 * 0) \geq 5 * 0 + 5
• 2(4 + 0) \geq 0 + 5
• 2(4) \geq 5
• 8 \geq 5

This is also true, confirming that x = 0 is also part of the solution set. Interpreting the solution correctly is crucial for understanding the implications of the inequality and applying it to real-world problems. The solution x ${\leq}$ 3 represents an infinite set of values, all of which satisfy the given condition.

Step 4: Verification and the Importance of Checking Solutions

Before declaring our solution as final, it's always a good practice to verify it. This step ensures that we haven't made any errors during the simplification process. To verify the solution, we can pick a value within the solution set (x ${\leq}$ 3) and a value outside the solution set (x > 3) and substitute them into the original inequality. Let's start by choosing a value within the solution set, say x = 2. Substituting this into the original inequality $2(4 + 2x) \geq 5x + 5$, we get:

• 2(4 + 2 * 2) \geq 5 * 2 + 5
• 2(4 + 4) \geq 10 + 5
• 2(8) \geq 15
• 16 \geq 15

This is true, which supports our solution. Now, let's choose a value outside the solution set, say x = 4. Substituting this into the original inequality, we get:

• 2(4 + 2 * 4) \geq 5 * 4 + 5
• 2(4 + 8) \geq 20 + 5
• 2(12) \geq 25
• 24 \geq 25

This is false, which further confirms that our solution x ${\leq}$ 3 is correct. Verification is a crucial step in problem-solving because it helps to catch mistakes and build confidence in the final answer. It also reinforces the understanding of the inequality and its solution set. By checking our solution with different values, we can be sure that we have accurately solved the problem.

Conclusion: The Solution and Its Implications

In conclusion, by following a step-by-step approach, we have successfully solved the inequality $2(4+2x) \geq 5x+5$. We began by distributing the multiplication, then combined like terms to isolate the variable x. This led us to the solution x ${\leq}$ 3, which means that any value of x less than or equal to 3 will satisfy the original inequality. We also emphasized the importance of verifying the solution by substituting values both within and outside the solution set into the original inequality. This process not only confirms the correctness of our answer but also deepens our understanding of the inequality and its implications.

Solving inequalities is a fundamental skill in mathematics, with applications in various fields such as economics, physics, and computer science. Understanding the principles and techniques involved in solving inequalities allows us to model and analyze real-world problems effectively. The ability to manipulate inequalities and interpret their solutions is crucial for making informed decisions and solving complex problems. This article has provided a comprehensive guide to solving the specific inequality $2(4+2x) \geq 5x+5$, but the principles and techniques discussed can be applied to a wide range of inequality problems. By mastering these skills, you can enhance your mathematical abilities and confidently tackle more challenging problems in the future. Therefore, the correct answer to the inequality is x ${\leq}$ 3, which corresponds to option C. This step-by-step solution not only provides the answer but also equips you with the knowledge and understanding to solve similar problems effectively.