Matching Solutions To Inequalities A Step By Step Guide

Hey guys! Today, we're diving into the world of inequalities and how to match them with their correct solutions. It might sound a bit daunting at first, but trust me, it's super manageable once you get the hang of it. We'll break down each inequality step-by-step, so you can confidently match them to their solutions. Let’s get started!

Understanding Inequalities

First off, what exactly is an inequality? In simple terms, an inequality is a mathematical statement that compares two expressions using symbols like >, <, ≥, or ≤. These symbols mean 'greater than,' 'less than,' 'greater than or equal to,' and 'less than or equal to,' respectively. Unlike equations, which have one specific solution, inequalities often have a range of solutions. For example, x > 5 means that x can be any number greater than 5, not just one particular number.

Why are inequalities important? Well, in real life, we rarely encounter situations where things are exactly equal. Think about speed limits on a road (you can drive at or below the limit), budgeting (you need to spend less than or equal to your income), or even cooking (you might need 'at least' a certain amount of an ingredient). Inequalities help us model and solve these real-world scenarios. When we talk about solving inequalities, we're essentially finding the range of values that make the inequality true. This range is often represented graphically on a number line or expressed in interval notation. Understanding how to manipulate and solve inequalities is a crucial skill in algebra and beyond. So, grab your thinking caps, and let’s jump into solving some inequalities and matching them to their solutions!

Basic Principles for Solving Inequalities

Before we dive into matching inequalities to their solutions, let’s quickly review some key principles for solving them. Solving inequalities is quite similar to solving equations, but there’s one crucial difference: when you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number reverses the order of the numbers on the number line. For instance, while 2 < 3, multiplying both sides by -1 gives -2 > -3.

Here are the basic principles:

1. Addition and Subtraction: You can add or subtract the same number from both sides of an inequality without changing the direction of the inequality sign. For example, if x + 3 > 5, you can subtract 3 from both sides to get x > 2.
2. Multiplication and Division by a Positive Number: You can multiply or divide both sides of an inequality by the same positive number without changing the direction of the inequality sign. For example, if 2x < 8, you can divide both sides by 2 to get x < 4.
3. Multiplication and Division by a Negative Number: This is the tricky one! When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. For example, if -3x ≥ 9, you divide both sides by -3 and flip the sign to get x ≤ -3.
4. Simplifying: Always simplify both sides of the inequality before you start isolating the variable. This might involve combining like terms or using the distributive property.
5. Checking Your Solution: It’s always a good idea to check your solution by plugging a value from your solution range back into the original inequality to make sure it holds true.

With these principles in mind, let's tackle the inequalities and match them up!

Matching Inequalities to Solutions

Now, let’s get to the main task: matching each inequality to its correct solution. We'll take each inequality one by one, solve it, and then see which solution it corresponds to. This is where the fun begins, guys! We've got four inequalities to crack, so let's roll up our sleeves and get to work. Remember, the goal is not just to find the answer, but to understand the process. So, let's make sure we're clear on each step as we go along. Okay, team, let's do this!

Inequality 1: $-3x > -36$

Our first inequality is -3x > -36. To solve for x, we need to isolate x on one side of the inequality. Remember that key rule? We're going to be dividing both sides by a negative number, so what do we need to do? That's right, we need to flip the inequality sign! Let's do it:

1. Divide both sides by -3: (-3x) / -3 < (-36) / -3 (Notice how the > sign flipped to <).
2. Simplify: This gives us x < 12.

So, the solution to the first inequality is x < 12. This means x can be any number less than 12, but not equal to 12. Keep this solution in mind as we move on to the next inequality.

Inequality 2: $b + 5 > 23$

Next up, we have the inequality b + 5 > 23. This one looks a bit simpler, right? Our goal here is to isolate b. To do that, we need to get rid of the +5 on the left side. What operation do we use to undo addition? Subtraction, of course! Let's go through the steps:

1. Subtract 5 from both sides: b + 5 - 5 > 23 - 5
2. Simplify: This leaves us with b > 18.

So, the solution to the second inequality is b > 18. This means b can be any number greater than 18. Notice that we didn’t flip the inequality sign here because we subtracted, not divided or multiplied by a negative number. Easy peasy, right? Let’s keep this solution in mind too.

Inequality 3: $1 + 7n ext{ } extgreater{=} ext{ } -90$

Alright, let's tackle the third inequality: 1 + 7n ≥ -90. This one involves a couple of steps, but don't worry, we've got this! We need to isolate n, so we'll first deal with the +1 and then the 7 that's multiplying n. Here’s how we'll break it down:

1. Subtract 1 from both sides: 1 + 7n - 1 ≥ -90 - 1
2. Simplify: This gives us 7n ≥ -91
3. Divide both sides by 7: (7n) / 7 ≥ (-91) / 7
4. Simplify: Finally, we get n ≥ -13

So, the solution to the third inequality is n ≥ -13. This means n can be any number greater than or equal to -13. Remember, the