Solving Linear Inequality: Which Ordered Pairs Work?

by Andrew McMorgan 53 views

Hey guys! Let's dive into the world of linear inequalities and figure out which ordered pairs actually make them true. It might sound a bit intimidating, but trust me, it's totally doable. We're going to break down the process step by step, so you'll be a pro in no time. We'll focus on the inequality 2x + y ≥ -5 and explore how to determine if a given ordered pair (x, y) is a solution. So, grab your thinking caps, and let's get started!

Understanding Linear Inequalities

Okay, so what exactly is a linear inequality? Think of it as a cousin of a linear equation, but instead of an equals sign (=), we've got inequality symbols like greater than (>) , less than (<), greater than or equal to (≥), or less than or equal to (≤). These symbols tell us that the relationship between the expressions on either side isn't necessarily an exact equality, but rather a range of possible values. Linear inequalities, just like their equation counterparts, represent a straight line when graphed, but the solutions to the inequality aren't just the points on the line. Instead, they include an entire region of the coordinate plane. This region is called the solution set, and it's made up of all the ordered pairs (x, y) that make the inequality true. This is a crucial concept to grasp before we start plugging in ordered pairs. Imagine the line dividing the coordinate plane into two halves. One of these halves (and sometimes the line itself) will be the solution set. Our mission is to figure out which half it is! So, how do we do that? That's where ordered pairs come in.

Key Concepts to Remember

Before we jump into solving, let’s solidify some key concepts:

  • Ordered Pair: An ordered pair (x, y) represents a point on the coordinate plane. The first value, x, is the x-coordinate, and the second value, y, is the y-coordinate.
  • Linear Inequality: A mathematical statement that compares two expressions using inequality symbols (>, <, ≥, ≤).
  • Solution Set: The set of all ordered pairs that make the inequality true.
  • Graphing Linear Inequalities: The solution set of a linear inequality can be represented graphically as a shaded region on the coordinate plane. The boundary line is solid for ≥ and ≤ (inclusive) and dashed for > and < (exclusive).

Understanding these concepts is essential for tackling problems involving linear inequalities. It's like having the right tools in your toolbox before starting a DIY project. Now that we've got our tools, let's see how to use them!

How to Check Ordered Pairs

The core of this problem lies in knowing how to check if an ordered pair is a solution to a given inequality. The process is actually pretty straightforward: we simply substitute the x and y values from the ordered pair into the inequality and see if the resulting statement is true. Let's break it down step-by-step:

  1. Identify the Inequality: First, make sure you know the inequality you're working with. In our case, it's 2x + y ≥ -5.
  2. Get the Ordered Pair: You'll be given an ordered pair, like (1, -2). Remember, the first number is the x-value, and the second is the y-value.
  3. Substitute: Replace the x and y in the inequality with the corresponding values from the ordered pair. So, for (1, -2), we'd have 2(1) + (-2) ≥ -5.
  4. Simplify: Do the math! Calculate the left side of the inequality. In our example, 2(1) + (-2) simplifies to 2 - 2 = 0.
  5. Check the Statement: Now we have 0 ≥ -5. Is this true? Yes, it is! Zero is indeed greater than or equal to negative five.
  6. Conclusion: If the statement is true, the ordered pair is a solution to the inequality. If it's false, the ordered pair is not a solution. Since 0 ≥ -5 is true, the ordered pair (1, -2) is a solution to the inequality 2x + y ≥ -5.

Let's try another example to really nail this down. Suppose we want to check the ordered pair (-3, 4). Substituting, we get 2(-3) + 4 ≥ -5, which simplifies to -6 + 4 ≥ -5, and further to -2 ≥ -5. This statement is also true, since -2 is greater than -5. So, (-3, 4) is another solution. But what if we tried (-2, -10)? Substituting, we get 2(-2) + (-10) ≥ -5, which simplifies to -4 - 10 ≥ -5, and then to -14 ≥ -5. This statement is false. -14 is definitely not greater than or equal to -5. Therefore, (-2, -10) is not a solution to the inequality. See? Once you get the hang of the substitution and simplification, it's just a matter of checking if the resulting statement makes sense.

Applying the Method to 2x + y ≥ -5

Alright, now let's put this method into action with our specific inequality: 2x + y ≥ -5. We're going to walk through a few examples to show you how to determine which ordered pairs are solutions. We'll choose a variety of pairs, some that work and some that don't, so you can see the process in action.

Example 1: Ordered Pair (0, 0)

This is a classic one to start with! Let's substitute x = 0 and y = 0 into our inequality:

2(0) + (0) ≥ -5

Simplifies to:

0 + 0 ≥ -5

Which is:

0 ≥ -5

Is this true? Yes! Zero is greater than -5. So, the ordered pair (0, 0) is a solution to the inequality. This means that the origin lies within the solution region when we graph the inequality.

Example 2: Ordered Pair (-2, -1)

Let's see if (-2, -1) works:

2(-2) + (-1) ≥ -5

Simplifies to:

-4 - 1 ≥ -5

Which is:

-5 ≥ -5

Is this true? Yes! -5 is equal to -5. Remember, the ≥ symbol means