Inequality Solution Set Does -25 Belong? A Step-by-Step Analysis

In the realm of mathematics, inequalities play a crucial role in defining relationships between values. An inequality is a statement that compares two expressions using inequality symbols such as >, <, ≤, or ≥. Solving inequalities involves finding the range of values that satisfy the given condition. This article delves into the process of determining which inequality from a given set includes -25 within its solution set. We will explore each inequality step-by-step, providing a clear and concise explanation to enhance understanding. Our main keyword, “inequality solution set,” will be central to our discussion, ensuring that readers grasp the core concept. We will focus on four specific inequalities, examining how to isolate the variable y and interpret the results. Whether you're a student grappling with algebra or simply looking to refresh your mathematical knowledge, this guide offers valuable insights and practical techniques for solving inequalities and identifying solution sets.

Understanding Inequality Solution Sets

Before we dive into the specific inequalities, it's essential to grasp the fundamental concept of an inequality solution set. The solution set of an inequality comprises all the values that, when substituted for the variable, make the inequality true. Unlike equations, which often have a single solution or a finite set of solutions, inequalities typically have an infinite number of solutions. This is because the solution is a range of values rather than a single point. For instance, the inequality x > 5 has a solution set that includes all numbers greater than 5, such as 5.001, 6, 10, 100, and so on. Representing these solutions can be done in several ways, including interval notation, graphing on a number line, or simply stating the condition in words. Understanding how to identify and represent these solution sets is crucial for solving more complex problems and applying inequalities in various real-world scenarios. The keyword inequality solution set is pivotal here as we aim to clarify this concept thoroughly.

Analyzing the First Inequality: y + 11 > -9

The first inequality we'll examine is y + 11 > -9. To determine if -25 is in the solution set, we need to isolate the variable y. This involves performing inverse operations to undo the addition. In this case, we subtract 11 from both sides of the inequality. This maintains the balance of the inequality, ensuring that the relationship between the two sides remains valid. The step-by-step process is as follows:

• Original inequality: y + 11 > -9
• Subtract 11 from both sides: y + 11 - 11 > -9 - 11
• Simplify: y > -20

Now we have the simplified inequality y > -20. This inequality states that y must be greater than -20. To check if -25 is in the solution set, we compare -25 to -20. Since -25 is less than -20, it does not satisfy the condition y > -20. Therefore, -25 is not in the solution set for this inequality. Understanding this process is crucial for grasping how to manipulate inequalities and determine their solution sets. The concept of an inequality solution set is directly applied here, as we are identifying whether a specific value fits within the range defined by the inequality.

Evaluating the Second Inequality: y - 17 ≤ -46

Next, we consider the inequality y - 17 ≤ -46. Similar to the previous example, our goal is to isolate the variable y to determine the solution set. In this case, we have subtraction, so we perform the inverse operation, which is addition. We add 17 to both sides of the inequality to isolate y. This keeps the inequality balanced and preserves the relationship between the two sides. The steps are as follows:

• Original inequality: y - 17 ≤ -46
• Add 17 to both sides: y - 17 + 17 ≤ -46 + 17
• Simplify: y ≤ -29

The simplified inequality is y ≤ -29. This means that y must be less than or equal to -29. Now we check if -25 is in the solution set. Since -25 is greater than -29, it does not satisfy the condition y ≤ -29. Thus, -25 is not part of the solution set for this inequality. This process illustrates how different inequality symbols affect the range of acceptable solutions. The inequality solution set concept is further reinforced as we identify whether a given value meets the criteria set by the inequality.

Investigating the Third Inequality: 30 ≤ y + 5

Now let's analyze the inequality 30 ≤ y + 5. To isolate y, we need to subtract 5 from both sides of the inequality. This maintains the balance and helps us determine the solution set. The steps are as follows:

• Original inequality: 30 ≤ y + 5
• Subtract 5 from both sides: 30 - 5 ≤ y + 5 - 5
• Simplify: 25 ≤ y

The simplified inequality is 25 ≤ y, which can also be written as y ≥ 25. This means that y must be greater than or equal to 25. To check if -25 is in the solution set, we compare -25 to 25. Since -25 is not greater than or equal to 25, it does not satisfy the condition. Therefore, -25 is not in the solution set for this inequality. This example highlights how rewriting an inequality can sometimes make it easier to interpret and understand the solution set. Our focus on the inequality solution set remains consistent as we evaluate whether a specific value meets the inequality's criteria.

Examining the Fourth Inequality: y + 20 > -30

Finally, we examine the inequality y + 20 > -30. To isolate y, we need to subtract 20 from both sides of the inequality. This keeps the inequality balanced and allows us to find the solution set. Here are the steps:

• Original inequality: y + 20 > -30
• Subtract 20 from both sides: y + 20 - 20 > -30 - 20
• Simplify: y > -50

The simplified inequality is y > -50. This means that y must be greater than -50. To determine if -25 is in the solution set, we compare -25 to -50. Since -25 is greater than -50, it satisfies the condition y > -50. Therefore, -25 is in the solution set for this inequality. This example clearly demonstrates how to identify a value within the solution set of an inequality. The concept of inequality solution set is central here, as we confirm that -25 meets the criteria defined by the inequality.

In conclusion, by systematically solving each inequality, we determined that -25 is in the solution set only for the inequality y + 20 > -30. This process involved isolating the variable y in each inequality and comparing -25 to the resulting solution set. Understanding how to manipulate inequalities and interpret their solution sets is a fundamental skill in mathematics. The inequality solution set is a crucial concept that underlies this process, allowing us to identify the range of values that satisfy a given condition. Through this detailed analysis, we hope to have provided a clear and comprehensive guide to solving inequalities and determining their solution sets, making it easier for students and math enthusiasts alike to tackle similar problems in the future.