Solving For Y In Inequalities A Step-by-Step Guide

In mathematics, solving for a variable in an inequality is a fundamental skill. This article will guide you through the process of solving the linear inequality $-65y + 19 < -2y + 41$ for $y$. We will break down each step, ensuring you understand the logic behind the operations. This detailed explanation will not only help you solve this specific problem but also equip you with the knowledge to tackle similar inequalities with confidence. Understanding how to manipulate inequalities is crucial for various mathematical concepts and real-world applications. By the end of this guide, you'll have a solid grasp of the techniques involved, including isolating variables, combining like terms, and dealing with negative coefficients. Let's embark on this journey to master the art of solving linear inequalities.

Understanding Linear Inequalities

Before we dive into the solution, let's first understand what linear inequalities are. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). These inequalities define a range of values that satisfy the given condition, unlike equations that have specific solutions. For example, the inequality $x > 5$ means that $x$ can be any value greater than 5, but not 5 itself. Linear inequalities are prevalent in various fields, including economics, physics, and computer science, where constraints and ranges are crucial. Understanding how to manipulate and solve these inequalities is a fundamental skill in mathematics and its applications. The ability to solve linear inequalities allows us to model real-world scenarios with constraints and find solutions that satisfy those constraints. By mastering the techniques of solving linear inequalities, we can gain valuable insights and make informed decisions in various situations. Let's delve deeper into the steps required to solve linear inequalities and tackle the problem at hand with confidence.

Step-by-Step Solution

To solve the inequality $-65y + 19 < -2y + 41$, we will follow a series of algebraic steps to isolate $y$ on one side of the inequality. The key is to perform operations that maintain the balance of the inequality, ensuring that the relationship between the two sides remains consistent. This involves adding or subtracting terms from both sides, multiplying or dividing both sides by a constant, and simplifying the expressions. Throughout the process, we must pay close attention to the direction of the inequality sign, especially when multiplying or dividing by a negative number. By carefully executing each step, we can systematically isolate the variable and determine the range of values that satisfy the inequality. Let's begin by grouping the $y$ terms on one side and the constant terms on the other side. This will help us simplify the inequality and move closer to isolating the variable. Remember, the goal is to get $y$ by itself on one side of the inequality, so we can determine the values of $y$ that make the inequality true.

1. Grouping $y$ terms

Our first step is to group the terms containing $y$ on one side of the inequality. To do this, we will add $65y$ to both sides of the inequality. This operation will eliminate the $-65y$ term on the left side, effectively moving the $y$ terms to the right side. Adding the same value to both sides ensures that the inequality remains balanced and the relationship between the two sides is preserved. This is a crucial step in isolating the variable and simplifying the inequality. By grouping the $y$ terms, we can combine them and work towards isolating $y$ on one side. The resulting inequality will be easier to manipulate and solve. Let's proceed with this step and see how it transforms the original inequality.

$-65y + 19 < -2y + 41$

Add $65y$ to both sides:

$-65y + 19 + 65y < -2y + 41 + 65y$

This simplifies to:

$19 < 63y + 41$

2. Grouping Constant Terms

Next, we need to group the constant terms on the other side of the inequality. Currently, we have $19$ on the left side and $+41$ on the right side. To move the constant term to the left side, we will subtract $41$ from both sides of the inequality. This operation will isolate the term with $y$ on the right side. Subtracting the same value from both sides maintains the balance of the inequality and ensures that the relationship between the two sides remains consistent. This step is essential in isolating the variable and simplifying the inequality further. By grouping the constant terms, we can consolidate them and work towards isolating $y$ on one side. The resulting inequality will be even closer to the desired form, where $y$ is by itself on one side. Let's proceed with this step and observe how it transforms the inequality.

$19 < 63y + 41$

Subtract $41$ from both sides:

$19 - 41 < 63y + 41 - 41$

This simplifies to:

$-22 < 63y$

3. Isolate $y$

Now, we need to isolate $y$ completely. We have the inequality $-22 < 63y$. To get $y$ by itself, we need to divide both sides of the inequality by the coefficient of $y$, which is $63$. This operation will isolate $y$ on the right side and give us the solution to the inequality. Dividing both sides by the same positive value maintains the direction of the inequality. If we were dividing by a negative number, we would need to flip the inequality sign. This step is crucial in determining the range of values for $y$ that satisfy the inequality. By dividing both sides by the coefficient of $y$, we can find the values of $y$ that make the inequality true. Let's proceed with this step and find the solution for $y$.

$-22 < 63y$

Divide both sides by $63$:

rac{-22}{63} < rac{63y}{63}

This simplifies to:

rac{-22}{63} < y

4. Rewrite the Solution

The solution to the inequality is rac{-22}{63} < y. This means that $y$ is greater than -rac{22}{63}. We can also write this as y > -rac{22}{63}. This form is often preferred as it clearly states the values that $y$ can take. The fraction -rac{22}{63} is already in its simplest form, as 22 and 63 have no common factors other than 1. Therefore, we don't need to reduce the fraction further. The solution y > -rac{22}{63} represents all values of $y$ that are greater than -rac{22}{63}. This range of values satisfies the original inequality. Understanding how to express the solution in different forms is essential for interpreting the results and communicating them effectively. The solution y > -rac{22}{63} provides a clear and concise answer to the problem.

Final Answer

The solution to the inequality $-65y + 19 < -2y + 41$ is:

y > -rac{22}{63}

This means that any value of $y$ greater than -rac{22}{63} will satisfy the original inequality. We have successfully solved for $y$ by following a step-by-step process of grouping terms and isolating the variable. This solution provides a clear understanding of the range of values that make the inequality true. By mastering the techniques used in this solution, you can confidently tackle similar linear inequalities and solve for variables in various mathematical problems.

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