Solving Inequalities Step By Step Guide To 0.2(x+20)-3 > -7-6.2x

In the realm of mathematics, inequalities play a crucial role in defining relationships between values that are not necessarily equal. Unlike equations, which seek to find specific solutions, inequalities describe a range of possible solutions. Understanding how to solve inequalities is fundamental for various mathematical applications, from optimizing real-world scenarios to understanding the behavior of functions. This article will provide a comprehensive, step-by-step guide to solving the inequality $0.2(x+20)-3 > -7-6.2x$, ensuring that readers of all levels can grasp the concepts and techniques involved. We will break down each step, explaining the underlying principles and providing clear, concise explanations. By the end of this article, you will not only be able to solve this specific inequality but also gain the confidence to tackle similar problems with ease. So, let's dive in and unravel the intricacies of solving inequalities!

1. Applying the Distributive Property

The first step in solving the inequality $0.2(x+20)-3 > -7-6.2x$ involves simplifying the expression by applying the distributive property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within parentheses. In simpler terms, it states that $a(b + c) = ab + ac$. This property is essential for expanding expressions and making them easier to work with. In our case, we need to distribute the $0.2$ across the terms inside the parentheses $(x+20)$. This means we multiply $0.2$ by both $x$ and $20$.

When we multiply $0.2$ by $x$, we get $0.2x$. This is a straightforward application of multiplication in algebra. Next, we multiply $0.2$ by $20$. To do this, we can think of $0.2$ as $\frac{2}{10}$, so we are essentially calculating $\frac{2}{10} \times 20$. This can be simplified to $2 \times \frac{20}{10}$, which equals $2 \times 2$, giving us $4$. Therefore, $0.2$ multiplied by $20$ equals $4$. Now that we have distributed the $0.2$, our expression becomes $0.2x + 4$. We still have the $-3$ term from the original inequality, so we bring that down to complete this part of the simplification. Thus, applying the distributive property, we transform the left side of the inequality into $0.2x + 4 - 3$. This step is crucial because it removes the parentheses, making it easier to combine like terms in the next step. Understanding and correctly applying the distributive property is a cornerstone of algebraic manipulation, and it's a skill that will be used repeatedly in more complex mathematical problems. This meticulous approach to simplification is what sets the stage for solving the inequality effectively.

2. Combining Like Terms

After applying the distributive property in the previous step, our inequality now looks like this: $0.2x + 4 - 3 > -7 - 6.2x$. The next logical step in simplifying the inequality is to combine like terms. Combining like terms involves identifying terms that have the same variable and exponent, or are constants, and then adding or subtracting them. This process helps to consolidate the expression, making it more manageable and easier to solve. In our case, we need to focus on the left side of the inequality, where we have the terms $4$ and $-3$. These are both constant terms, meaning they don't have any variables attached to them. Therefore, they can be combined.

To combine $4$ and $-3$, we simply perform the operation $4 - 3$, which equals $1$. So, the constant terms on the left side of the inequality simplify to $1$. The term $0.2x$ remains unchanged because there are no other terms on the left side that contain the variable $x$. After combining the like terms, the left side of the inequality becomes $0.2x + 1$. This simplification is a key step because it reduces the number of terms in the inequality, making it less complex. The inequality now reads $0.2x + 1 > -7 - 6.2x$. This simplified form allows us to proceed with the next steps in solving the inequality, which will involve isolating the variable $x$ on one side of the inequality. Combining like terms is a fundamental algebraic skill that is essential for solving not only inequalities but also equations and other mathematical expressions. By mastering this skill, you lay a solid foundation for tackling more advanced algebraic problems.

3. Utilizing the Addition Property of Inequality

Having simplified the inequality to $0.2x + 1 > -7 - 6.2x$, the next crucial step is to isolate the variable $x$ on one side of the inequality. To achieve this, we employ the addition property of inequality. This property states that adding the same value to both sides of an inequality does not change the validity or direction of the inequality. In simpler terms, if $a > b$, then $a + c > b + c$. This principle is fundamental for manipulating inequalities while preserving their integrity. Our goal here is to gather all the terms containing $x$ on one side, typically the left side, and all the constant terms on the other side. Looking at our inequality, we have $-6.2x$ on the right side. To eliminate this term from the right side and move it to the left, we need to add its opposite, which is $+6.2x$, to both sides of the inequality.

When we add $6.2x$ to both sides, we get: $(0.2x + 1) + 6.2x > (-7 - 6.2x) + 6.2x$. Now, we simplify each side separately. On the left side, we combine the like terms $0.2x$ and $6.2x$. Adding these together, we get $0.2x + 6.2x = 6.4x$. So, the left side of the inequality becomes $6.4x + 1$. On the right side, we have $-6.2x + 6.2x$, which cancels out to zero, leaving us with just $-7$. Therefore, the right side of the inequality simplifies to $-7$. After applying the addition property and simplifying, our inequality now looks like this: $6.4x + 1 > -7$. This transformation is a significant step forward because it has consolidated all the $x$ terms on one side, making it easier to proceed with isolating $x$. The addition property of inequality is a powerful tool that allows us to manipulate inequalities in a controlled manner, and it’s essential for solving a wide range of inequality problems.

4. Isolating the Variable Using Subtraction

Following the application of the addition property, our inequality is now in the form $6.4x + 1 > -7$. The next step towards solving for $x$ involves isolating the term with the variable on one side of the inequality. To do this, we need to eliminate the constant term, which is $+1$, from the left side. We can achieve this by using the subtraction property of inequality. Similar to the addition property, the subtraction property states that subtracting the same value from both sides of an inequality does not change the validity or direction of the inequality. If $a > b$, then $a - c > b - c$. This property is the counterpart to addition and is equally crucial for manipulating inequalities. In our case, we want to subtract $1$ from both sides of the inequality. This will effectively remove the $+1$ from the left side, bringing us closer to isolating $x$.

Subtracting $1$ from both sides of the inequality, we get: $(6.4x + 1) - 1 > -7 - 1$. Now, we simplify each side separately. On the left side, we have $1 - 1$, which equals zero. This leaves us with just $6.4x$ on the left side. On the right side, we have $-7 - 1$, which equals $-8$. Therefore, after subtracting $1$ from both sides and simplifying, our inequality becomes $6.4x > -8$. This simplified form is a significant milestone in solving the inequality. We have successfully isolated the term with $x$ on one side, and now we only need one more step to find the solution for $x$. The subtraction property of inequality is a fundamental tool in algebra, and mastering its use is essential for solving various types of equations and inequalities. By carefully applying this property, we maintain the balance of the inequality while moving closer to our goal of finding the solution.

5. Solving for x Using Division

Having reached the simplified form of the inequality, $6.4x > -8$, the final step in solving for $x$ involves using the division property of inequality. This property states that dividing both sides of an inequality by the same positive number does not change the direction of the inequality. However, if we divide by a negative number, the direction of the inequality must be reversed. In simpler terms, if $a > b$ and $c$ is positive, then $\frac{a}{c} > \frac{b}{c}$. But, if $c$ is negative, then $\frac{a}{c} < \frac{b}{c}$. This distinction is crucial and is a common point of error for students learning to solve inequalities. In our case, we need to divide both sides of the inequality by $6.4$ to isolate $x$. Since $6.4$ is a positive number, we do not need to reverse the direction of the inequality.

Dividing both sides of the inequality by $6.4$, we get: $\frac{6.4x}{6.4} > \frac{-8}{6.4}$. Now, we simplify each side. On the left side, $\frac{6.4x}{6.4}$ simplifies to $x$, as the $6.4$ in the numerator and denominator cancel each other out. On the right side, we have $\frac{-8}{6.4}$. To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor, or we can convert the decimal to a fraction. Let's convert $6.4$ to a fraction: $6.4 = \frac{64}{10}$. So, $\frac{-8}{6.4}$ becomes $\frac{-8}{\frac{64}{10}}$. To divide by a fraction, we multiply by its reciprocal, so we have $-8 \times \frac{10}{64}$. This simplifies to $\frac{-80}{64}$. Now, we can reduce this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $16$. So, $\frac{-80}{64}$ simplifies to $\frac{-5}{4}$, which is equal to $-1.25$. Therefore, after dividing both sides by $6.4$ and simplifying, our inequality becomes $x > -1.25$. This is the final solution to the inequality. It tells us that $x$ can be any number greater than $-1.25$. The division property of inequality is a critical tool for solving inequalities, and it’s important to remember the rule about reversing the inequality sign when dividing by a negative number. By correctly applying this property, we can successfully isolate the variable and find the solution set for the inequality.

Conclusion

In this comprehensive guide, we have meticulously walked through the steps to solve the inequality $0.2(x+20)-3 > -7-6.2x$. Starting with the distributive property, we simplified the expression by multiplying $0.2$ across the terms in parentheses. Next, we combined like terms to consolidate the inequality, making it more manageable. We then applied the addition property of inequality to gather the variable terms on one side and the constant terms on the other. Following this, we used the subtraction property of inequality to further isolate the variable term. Finally, we employed the division property of inequality to solve for $x$, ensuring we reversed the inequality sign when dividing by a negative number. The solution we arrived at is $x > -1.25$, which means that any value of $x$ greater than $-1.25$ will satisfy the original inequality. Solving inequalities is a fundamental skill in mathematics, and the steps we've outlined here provide a solid framework for tackling a wide range of inequality problems. By understanding and practicing these techniques, you can build confidence in your ability to solve algebraic problems effectively. Remember, the key to mastering mathematics is consistent practice and a clear understanding of the underlying principles. So, continue to explore and challenge yourself with new problems, and you'll find your mathematical skills growing stronger every day.