Solving The Inequality F + 11 ≤ 5/3 Interval Notation Solution
Introduction
In this article, we will delve into the process of solving the inequality f + 11 ≤ 5/3. Inequalities, a fundamental concept in mathematics, are mathematical statements that compare two expressions using symbols such as less than ( < ), greater than ( > ), less than or equal to ( ≤ ), and greater than or equal to ( ≥ ). Solving an inequality involves finding the range of values for the variable that makes the inequality true. Our focus here is on a linear inequality, which is an inequality where the variable's highest power is 1. This type of inequality is common in algebra and has practical applications in various fields, including economics, physics, and computer science. Understanding how to solve them is crucial for students and professionals alike. We will also explore how to express the solution in interval notation, a concise way of representing a set of numbers within a specific range. Interval notation uses brackets and parentheses to indicate whether the endpoints of the interval are included or excluded, providing a clear and efficient way to communicate the solution set of an inequality. By the end of this discussion, you should be able to confidently tackle similar problems and understand the underlying principles of inequality solutions.
Step-by-Step Solution
To solve the inequality f + 11 ≤ 5/3, we need to isolate the variable f on one side of the inequality. This involves performing algebraic operations on both sides of the inequality while maintaining its balance. The primary goal is to undo the operations that are being performed on f so that we can determine its possible values. The first step in this process is to subtract 11 from both sides of the inequality. Subtracting the same value from both sides ensures that the inequality remains valid and does not change the solution set. This step is based on the fundamental property of inequalities, which states that if a ≤ b, then a - c ≤ b - c for any real number c. By subtracting 11, we effectively remove the constant term from the left side, bringing us closer to isolating f. It’s crucial to perform this operation accurately, as any errors in this step will propagate through the rest of the solution. We'll then simplify the resulting expression by performing the subtraction on the right side. This involves finding a common denominator to combine the fraction and the integer, which is a fundamental skill in arithmetic and algebra. This simplification will give us a single numerical value on the right side, making it easier to interpret the solution in the subsequent steps. The arithmetic involved here is a crucial part of solving the inequality correctly and efficiently.
1. Subtract 11 from Both Sides
To isolate the variable f, we begin by subtracting 11 from both sides of the inequality:
f + 11 - 11 ≤ 5/3 - 11
Subtracting 11 from both sides is a valid algebraic manipulation because it preserves the inequality. This is a crucial step in isolating the variable and bringing us closer to the solution. The principle behind this step is that if we perform the same operation on both sides of an inequality, the relationship between the two sides remains unchanged. This allows us to simplify the inequality while maintaining its validity. Without this step, we would not be able to isolate f and determine the range of values that satisfy the inequality. The clarity and precision of this step are essential for arriving at the correct solution. It sets the foundation for the subsequent steps and ensures that the final answer is accurate.
2. Simplify the Right-Hand Side
To simplify the right-hand side, we need to express 11 as a fraction with a denominator of 3:
11 = 11/1 = (11 * 3) / (1 * 3) = 33/3
Now we can rewrite the inequality as:
f ≤ 5/3 - 33/3
Combining the fractions, we get:
f ≤ (5 - 33) / 3
f ≤ -28/3
This simplification step is critical because it consolidates the constant terms into a single fraction, making it easier to interpret the solution. Finding a common denominator is a fundamental skill in fraction arithmetic, and it is essential for combining fractions accurately. In this case, converting 11 to 33/3 allows us to subtract it from 5/3, resulting in a single fraction that represents the upper bound for f. The accuracy of this step directly impacts the final solution, so it is important to perform the arithmetic carefully. The simplified inequality, f ≤ -28/3, now clearly defines the range of values that f can take to satisfy the original inequality. This makes it easier to express the solution in interval notation, which is the next step in our process.
Expressing the Solution in Interval Notation
The solution to the inequality f ≤ -28/3 represents all real numbers less than or equal to -28/3. In interval notation, this is expressed as (-∞, -28/3]. Interval notation is a concise way of representing a set of numbers within a specific range. It uses parentheses and brackets to indicate whether the endpoints of the interval are included or excluded. A parenthesis indicates that the endpoint is not included, while a bracket indicates that the endpoint is included. The symbol ∞ (infinity) represents a quantity without any bound, and it is always enclosed in a parenthesis because infinity is not a specific number and cannot be included in an interval. Understanding interval notation is crucial for expressing solutions to inequalities and other mathematical problems involving sets of numbers. It provides a clear and efficient way to communicate the range of values that satisfy a given condition. The ability to convert between inequality notation and interval notation is a fundamental skill in algebra and calculus.
Understanding Interval Notation
The interval notation (-∞, -28/3] indicates that the solution includes all numbers from negative infinity up to -28/3, including -28/3 itself. The parenthesis next to -∞ signifies that negative infinity is not a specific number and is not included in the interval. The bracket next to -28/3 indicates that -28/3 is part of the solution set. This distinction between parentheses and brackets is crucial in interval notation, as it precisely defines the boundaries of the solution. The use of interval notation allows for a clear and unambiguous representation of the solution, which is particularly important when dealing with complex inequalities or systems of inequalities. It provides a standardized way to express the range of values that satisfy a given mathematical condition, making it easier to communicate mathematical concepts and solutions. The understanding of interval notation is a foundational skill for advanced mathematical studies and applications.
Conclusion
In summary, to solve the inequality f + 11 ≤ 5/3, we first subtracted 11 from both sides and then simplified the right-hand side to get f ≤ -28/3. Finally, we expressed the solution in interval notation as (-∞, -28/3]. This process demonstrates the systematic approach to solving linear inequalities, which involves isolating the variable and expressing the solution in a clear and concise manner. The steps we followed, including subtracting constants and simplifying fractions, are fundamental algebraic techniques that apply to a wide range of mathematical problems. The use of interval notation provides a standardized way to represent the solution set, making it easier to communicate the results. Understanding these concepts and techniques is essential for students and professionals in various fields, as inequalities and their solutions are used in modeling and solving real-world problems. This comprehensive approach ensures that the solution is not only mathematically correct but also clearly understandable and applicable in various contexts.
By mastering these steps, you can confidently solve similar inequalities and express their solutions effectively. The ability to solve inequalities is a valuable skill in mathematics and many other disciplines, providing a foundation for more advanced mathematical concepts and applications. The consistent application of algebraic principles and attention to detail are key to success in solving inequalities and interpreting their solutions. This article has provided a thorough guide to solving a specific linear inequality, but the principles and techniques discussed can be applied to a wide variety of similar problems. The understanding and practice of these skills will enhance your mathematical proficiency and problem-solving abilities.
