Solving The Inequality -9/8 Y - 1 > 5/6 Y + 3/8 A Step-by-Step Guide

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Inequalities, a fundamental concept in mathematics, represent relationships where two values are not necessarily equal. They are essential for modeling real-world scenarios involving constraints, ranges, and comparisons. In this comprehensive guide, we will embark on a journey to master the art of solving inequalities, focusing on the specific example of $-\frac{9}{8}y - 1 > \frac{5}{6}y + \frac{3}{8}$. By delving into the intricacies of this inequality, we will uncover the underlying principles and techniques that empower us to tackle a wide array of inequality problems.

Understanding Inequalities: Before we dive into the solution, let's first grasp the essence of inequalities. Unlike equations that assert equality between two expressions, inequalities express relationships 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 inequality symbols: >, <, ≥, and ≤. Inequalities play a pivotal role in various mathematical domains, including algebra, calculus, and linear programming, as well as in practical applications such as optimization problems and decision-making processes.

The Challenge at Hand: Our focus is on unraveling the inequality $-\frac{9}{8}y - 1 > \frac{5}{6}y + \frac{3}{8}$. This inequality presents a linear relationship between the variable 'y' and a set of constants. Our objective is to isolate 'y' on one side of the inequality, thereby determining the range of values for 'y' that satisfy the given condition. To accomplish this, we will employ a series of algebraic manipulations, adhering to the fundamental properties of inequalities.

Step 1: Eliminating Fractions - A Foundation for Clarity

Fractions can often complicate mathematical expressions, making them appear daunting. In the context of inequalities, fractions can obscure the underlying relationships and hinder our progress towards a solution. Therefore, our initial step involves eliminating these fractions to simplify the inequality and pave the way for a more streamlined approach. To achieve this, we will employ a technique known as finding the least common multiple (LCM). The LCM is the smallest number that is a multiple of all the denominators present in the inequality. In our case, the denominators are 8 and 6. The LCM of 8 and 6 is 24. By multiplying both sides of the inequality by 24, we effectively eliminate the fractions, transforming the inequality into a more manageable form. This strategic move simplifies the subsequent steps and allows us to focus on the core algebraic manipulations.

The LCM Unveiled: The LCM, in essence, acts as a common denominator that allows us to combine fractions seamlessly. By multiplying each term in the inequality by the LCM, we are essentially scaling the entire expression while preserving the fundamental relationship between the two sides. This process ensures that the fractions are eliminated without altering the solution set of the inequality. The LCM technique is a cornerstone of algebraic manipulation, applicable not only to inequalities but also to equations and other mathematical expressions involving fractions.

Applying the LCM to Our Inequality: Let's apply this technique to our inequality, $-\frac{9}{8}y - 1 > \frac{5}{6}y + \frac{3}{8}$. Multiplying both sides of the inequality by 24, we get:

24 * (-$\frac{9}{8}y - 1$) > 24 * ($\frac{5}{6}y + \frac{3}{8}$)

This simplifies to:

-27y - 24 > 20y + 9

Now, the inequality is free of fractions, making it easier to manipulate and solve.

Step 2: Gathering 'y' Terms - Isolating the Unknown

Having successfully eliminated fractions, our next objective is to consolidate all the terms containing the variable 'y' on one side of the inequality. This strategic move brings us closer to isolating 'y' and determining its solution set. To achieve this, we will employ the principle of adding or subtracting the same quantity from both sides of the inequality. This principle ensures that the fundamental relationship between the two sides remains unchanged while allowing us to rearrange the terms as desired. In our specific inequality, we will add 27y to both sides to group the 'y' terms on the right side of the inequality. This maneuver simplifies the expression and sets the stage for further isolation of 'y'.

The Power of Balance: The concept of adding or subtracting the same quantity from both sides of an inequality stems from the fundamental principle of maintaining balance. Just as a scale remains balanced when equal weights are added or removed from both sides, an inequality preserves its relationship when the same quantity is added or subtracted from both sides. This principle is a cornerstone of algebraic manipulation, ensuring that our transformations do not alter the solution set of the inequality.

Applying the Principle: In our inequality, -27y - 24 > 20y + 9, we will add 27y to both sides:

-27y - 24 + 27y > 20y + 9 + 27y

This simplifies to:

-24 > 47y + 9

Now, all the 'y' terms are grouped on the right side of the inequality, bringing us closer to isolating 'y'.

Step 3: Isolating 'y' - The Final Push

With the 'y' terms consolidated on one side, our next goal is to isolate 'y' completely. This involves removing any constants or coefficients that are associated with 'y'. To achieve this, we will employ a combination of addition, subtraction, multiplication, and division, while adhering to the golden rule of inequalities: when multiplying or dividing both sides by a negative number, we must flip the inequality sign. This rule is crucial to preserve the correct direction of the inequality relationship.

The Golden Rule Unveiled: The rule of flipping the inequality sign when multiplying or dividing by a negative number arises from the nature of negative numbers. When we multiply or divide by a negative number, we are essentially reversing the number line. This reversal necessitates flipping the inequality sign to maintain the accurate relationship between the two sides. For instance, if a > b, then multiplying both sides by -1 gives -a < -b. This rule is a critical aspect of solving inequalities and must be applied diligently to avoid errors.

Step-by-Step Isolation: Let's apply these principles to our inequality, -24 > 47y + 9. First, we subtract 9 from both sides:

-24 - 9 > 47y + 9 - 9

This simplifies to:

-33 > 47y

Next, we divide both sides by 47:

$\frac{-33}{47}$ > y

Step 4: Expressing the Solution - Clarity and Precision

We have successfully isolated 'y', but to present the solution in a clear and concise manner, we will rewrite the inequality with 'y' on the left side. This is a matter of convention and enhances the readability of the solution. Additionally, we will express the solution in both inequality notation and interval notation. Inequality notation directly states the range of values for 'y', while interval notation provides a more compact representation using parentheses and brackets.

Rewriting for Readability: Rewriting the inequality with 'y' on the left side simply involves flipping the inequality and the variable. So, -$\frac{33}{47}$ > y becomes:

y < -$\frac{33}{47}$

This form is more intuitive, as it directly states that 'y' is less than -$\frac{33}{47}$.

Inequality Notation: The solution in inequality notation is simply:

y < -$\frac{33}{47}$

This notation explicitly defines the range of values for 'y' that satisfy the inequality.

Interval Notation: Interval notation provides a concise way to represent the solution set. It uses parentheses and brackets to indicate whether the endpoints are included or excluded. In our case, since 'y' is strictly less than -$\frac{33}{47}$, we use a parenthesis to exclude the endpoint. The interval notation for our solution is:

(-∞, -$\frac{33}{47}$)

This notation signifies that 'y' can take any value from negative infinity up to, but not including, -$\frac{33}{47}$.

In this comprehensive guide, we have meticulously solved the inequality $-\frac{9}{8}y - 1 > \frac{5}{6}y + \frac{3}{8}$, demonstrating the core principles and techniques involved in solving inequalities. We began by eliminating fractions using the LCM, then grouped the 'y' terms on one side, and finally isolated 'y' by applying the golden rule of flipping the inequality sign when multiplying or dividing by a negative number. We expressed the solution in both inequality notation and interval notation, ensuring clarity and precision.

By mastering these techniques, you are well-equipped to tackle a wide range of inequality problems. Remember, inequalities are not merely abstract mathematical concepts; they are powerful tools for modeling real-world constraints, comparisons, and optimization problems. So, embrace the challenge, practice diligently, and unlock the power of inequalities!

Keywords: solving inequalities, algebraic manipulation, least common multiple (LCM), inequality notation, interval notation, linear inequalities, mathematical problem solving, algebraic equations, variable isolation, solution sets.

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