Which Values Are Solutions Of The Inequality 5 Y-8

Solving the Inequality: 5y - 8 > 0 – A Comprehensive Guide

This article will explore the solution to the inequality 5y - 8 > 0, providing a step-by-step guide suitable for students of all levels. We will cover the process of solving the inequality, interpreting the solution, and representing it graphically. Understanding inequalities is crucial for various mathematical applications, and this guide aims to provide a clear and comprehensive understanding of the process. We will also delve into the underlying mathematical principles and address common questions.

Understanding Inequalities

Before diving into the solution, let's refresh our understanding of inequalities. Unlike equations, which state that two expressions are equal, inequalities compare two expressions using symbols like:

• > (greater than)
• < (less than)
• ≥ (greater than or equal to)
• ≤ (less than or equal to)

Solving an inequality means finding the range of values for the variable (in this case, y) that make the inequality true. The solution is typically expressed as an interval or using set notation.

Solving the Inequality 5y - 8 > 0

Let's tackle the inequality 5y - 8 > 0. The goal is to isolate y on one side of the inequality sign. We'll achieve this using the same algebraic principles used for solving equations, with one important exception: when multiplying or dividing by a negative number, you must reverse the inequality sign.

Step 1: Add 8 to both sides:

Our first step is to add 8 to both sides of the inequality to move the constant term to the right side:

5y - 8 + 8 > 0 + 8

This simplifies to:

5y > 8

Step 2: Divide both sides by 5:

Next, we divide both sides by 5 to isolate y:

5y / 5 > 8 / 5

This gives us the solution:

y > 8/5

Or, as a decimal:

y > 1.6

Interpreting the Solution

The solution y > 1.6 means that any value of y greater than 1.6 will satisfy the inequality 5y - 8 > 0. This includes values like 1.7, 2, 10, 100, and so on. It's important to note that y cannot be equal to 1.6; it must be strictly greater.

Representing the Solution Graphically

The solution can be represented graphically on a number line. We would typically draw an open circle (or parenthesis) at 1.6 to indicate that 1.6 is not included in the solution, and then shade the region to the right of 1.6 to represent all values greater than 1.6.

     <-------------------|------------------->
          0             1.6                 ∞
               o========>

This visual representation clearly shows the range of values that satisfy the inequality.

Set Notation and Interval Notation

The solution can also be expressed using set notation and interval notation:

• Set Notation: {y ∈ ℝ | y > 1.6} This reads as "the set of all real numbers y such that y is greater than 1.6". ℝ represents the set of all real numbers.

• Interval Notation: (1.6, ∞) This notation uses parentheses to indicate that 1.6 is not included, and ∞ (infinity) to represent that the solution extends indefinitely to the right.

Mathematical Principles at Play

The solution relies on several fundamental mathematical principles:

• Addition Property of Inequality: Adding the same number to both sides of an inequality does not change the direction of the inequality sign.
• Division Property of Inequality: Dividing both sides of an inequality by the same positive number does not change the direction of the inequality sign. If you divide by a negative number, you must reverse the inequality sign.
• Transitive Property of Inequality: If a > b and b > c, then a > c. This property allows us to chain inequalities together.

Frequently Asked Questions (FAQ)

Q: What if the inequality was 5y - 8 ≥ 0?

A: The only difference would be that the solution would include 1.6. The solution would be y ≥ 1.6, represented graphically with a closed circle at 1.6. The interval notation would be [1.6, ∞).

Q: How would I solve a more complex inequality involving multiple variables?

A: Solving more complex inequalities requires applying the same principles systematically. You would need to use the properties of inequalities to isolate the variable you are interested in, paying close attention to the direction of the inequality sign when multiplying or dividing by negative numbers. Consider using techniques like factoring or the quadratic formula if needed.

Q: Can the solution to an inequality be an empty set?

A: Yes, some inequalities have no solution (the empty set, denoted by Ø or {}). For example, the inequality x > x + 1 has no solution because no number can be simultaneously greater than itself plus one.

Q: What is the significance of solving inequalities in real-world problems?

A: Inequalities are essential for modeling and solving real-world problems involving constraints or limitations. For example, in finance, inequalities are used to model budget constraints; in engineering, to analyze stress limits; and in optimization problems, to find the maximum or minimum values within specific boundaries.

Conclusion

Solving the inequality 5y - 8 > 0 involves applying basic algebraic principles, understanding the properties of inequalities, and carefully interpreting the solution. The solution, y > 1.6, can be expressed in various ways, including graphically, using set notation, and using interval notation. This process is fundamental to a deeper understanding of algebra and its applications across numerous fields. Remember to always double-check your steps and consider the implications of the inequality sign when dealing with negative numbers. Mastering these concepts is crucial for success in higher-level mathematics and its applications in diverse fields. This comprehensive explanation should equip you with the necessary tools to tackle similar problems with confidence.