Solving For Y: -5 > Y + (-2) Inequality Explained

by Alex Johnson 50 views

Let's dive into the world of inequalities and solve this problem step-by-step. Inequalities might seem intimidating at first, but with a little understanding, they're just as manageable as regular equations. This guide will walk you through the process of isolating 'y' in the inequality -5 > y + (-2). We'll break down each step, making it super easy to follow along. Whether you're a student tackling homework or just brushing up on your math skills, you're in the right place!

Understanding Inequalities

Before we jump into the specific problem, let's quickly recap what inequalities are all about. Unlike equations, which show equality between two expressions, inequalities show a relationship where one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. The symbols we use are:

  • > Greater than
  • < Less than
  • ≥ Greater than or equal to
  • ≤ Less than or equal to

When solving inequalities, our goal is similar to solving equations: we want to isolate the variable (in this case, 'y') on one side of the inequality. However, there's one crucial difference: if we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality sign. Keep this in mind as we proceed!

Why Solve Inequalities?

Understanding how to solve inequalities is more than just a classroom exercise; it's a fundamental skill with real-world applications. Inequalities help us model and solve problems where quantities aren't necessarily equal but have a defined relationship. Here are a few practical scenarios where inequalities come in handy:

  • Budgeting and Finance: Imagine you have a certain amount of money to spend. You can use inequalities to determine how much you can afford on different items while staying within your budget. For example, if you have $100 and want to buy a shirt and a pair of pants, you can set up an inequality to ensure the total cost doesn't exceed your budget.
  • Setting Limits and Constraints: Many situations involve constraints or limits. For instance, a roller coaster might have a minimum and maximum height requirement. These limits can be expressed and analyzed using inequalities.
  • Optimization Problems: In business and engineering, inequalities are used to optimize processes. For example, a company might want to maximize its profits while staying within certain resource constraints. Inequalities help define these constraints and find the optimal solution.
  • Scientific Modeling: In various scientific fields, inequalities are used to describe ranges of values or conditions. For example, in chemistry, inequalities can represent the pH range of a solution.
  • Everyday Decision Making: Even in our daily lives, we often use inequalities without realizing it. When deciding how much time to spend on different activities or how many items to buy, we're often considering constraints and making decisions based on inequalities.

In essence, inequalities provide a powerful tool for dealing with situations where exact equality isn't the primary concern. They allow us to work with ranges of values and establish boundaries, making them essential for problem-solving across various disciplines.

Breaking Down the Problem: -5 > y + (-2)

Now, let's get back to our specific inequality: -5 > y + (-2). Our mission is to isolate 'y' on one side of the inequality. Here’s how we'll do it:

  1. Simplify the inequality: The first thing we notice is the '+ (-2)'. Adding a negative number is the same as subtracting the positive version of that number. So, we can rewrite the inequality as: -5 > y - 2

    This simplification makes the inequality a bit easier to work with. We've eliminated the unnecessary plus and minus signs, making the next steps more straightforward.

  2. Isolate 'y': To get 'y' by itself, we need to get rid of the '- 2' on the right side of the inequality. We can do this by adding 2 to both sides. Remember, whatever operation we perform on one side of the inequality, we must perform on the other side to maintain the balance. So, we add 2 to both sides:

    -5 + 2 > y - 2 + 2

    This step is crucial because it begins to isolate 'y'. By adding 2 to both sides, we're effectively canceling out the '- 2' on the right side, bringing us closer to our goal.

  3. Simplify again: Now, let's simplify both sides of the inequality:

    -3 > y

    On the left side, -5 + 2 equals -3. On the right side, -2 + 2 cancels out, leaving us with just 'y'. This simplification gives us a clearer picture of the solution.

  4. Rewrite (optional but recommended): While -3 > y is a perfectly valid answer, it's often easier to understand if we rewrite it with 'y' on the left side. To do this, we simply flip the inequality. But remember, when we flip the inequality, we also need to flip the inequality sign. So, -3 > y becomes:

    y < -3

    This final step is about clarity. By having 'y' on the left side, we can read the solution more intuitively. It tells us directly that 'y' is less than -3.

Visualizing the Solution

It's helpful to visualize what our solution, y < -3, means. We can do this by drawing a number line. A number line is a visual representation of numbers, extending infinitely in both positive and negative directions.

  1. Draw a number line: Draw a straight line and mark zero (0) in the middle. Then, mark positive numbers to the right and negative numbers to the left. Include -3 on your number line.
  2. Mark the solution: Since y < -3, we're looking for all numbers less than -3. On the number line, this is represented by all the numbers to the left of -3.
  3. Use an open circle: Because y is strictly less than -3 (not less than or equal to), we use an open circle at -3. This indicates that -3 itself is not included in the solution.
  4. Draw an arrow: Draw an arrow extending to the left from the open circle, indicating that all numbers less than -3 are part of the solution.

This visual representation gives us a clear understanding of the solution set. It includes all numbers to the left of -3 on the number line, excluding -3 itself.

Expressing the Solution in Different Ways

Understanding that the solution y < -3 is just one way to express our answer. Here are a couple of other ways you might see the solution represented:

  • Interval Notation: Interval notation is a concise way to represent a set of numbers. For y < -3, the interval notation is (-∞, -3). The parenthesis indicates that -3 is not included in the solution, and -∞ represents negative infinity, indicating that the solution extends indefinitely to the left.

  • Set-Builder Notation: Set-builder notation uses mathematical symbols to describe the set of all numbers that satisfy a condition. For y < -3, the set-builder notation is {y | y < -3}. This is read as