Hey everyone! Today, we're diving into a cool math concept: inequalities. We'll be looking at a specific problem, but the skills you learn here can be applied to tons of other situations. So, let's break down the problem: "The sum of 13 and a number t is no more than 24." Sounds a bit formal, right? Don't worry, we'll translate this into something super easy to understand. This is like a puzzle, and we're going to figure out what numbers t can be. Get ready to flex those math muscles!
Decoding the Problem: What Does "No More Than" Mean?
Alright, first things first: let's understand the key phrase, "no more than." When you see this in a math problem, it's code for a specific inequality symbol. Think about it this way: if something is "no more than" 24, it can be 24, or it can be less than 24. It can't be bigger than 24. So, "no more than" translates to ≤ (less than or equal to). We also have to define the concept of t. The variable t is just a number we don't know yet.
So, we can start to translate the problem step by step. We know that the sum of 13 and t will be expressed mathematically as 13 + t. Because this sum is "no more than" 24, we can write the entire statement as an inequality, like this: 13 + t ≤ 24. Congrats, guys! You've now successfully converted a word problem into a mathematical inequality. This is like learning a new language – once you get the basics, everything starts to make sense. Now, let's see how we can solve this inequality and find all the possible values of t. This step is where we apply our algebra knowledge. It's very similar to solving equations.
Remember, inequalities are like balanced scales. Whatever you do to one side, you have to do to the other to keep things balanced. In this case, our goal is to isolate t on one side of the inequality. This is super easy; we need to get rid of the 13 that is currently being added to t. To do this, we subtract 13 from both sides of the inequality. This results in t on the left side and a new value on the right side. So, starting with 13 + t ≤ 24. Then we subtract 13 from both sides, so 13 + t - 13 ≤ 24 - 13. Thus, t ≤ 11. Boom! We've solved it! This means t can be any number that is less than or equal to 11.
Solving the Inequality: Finding the Range of t
Now that we have the inequality, let's solve it. Solving an inequality is very similar to solving an equation. Our goal is to isolate the variable t. In our case, we have the inequality: 13 + t ≤ 24. To isolate t, we need to get rid of the 13 that's being added to it. We do this by subtracting 13 from both sides of the inequality. Remember, whatever you do to one side, you must do to the other to keep things balanced. So, we get: 13 + t - 13 ≤ 24 - 13. This simplifies to t ≤ 11. That's it! We've solved the inequality. This means that t can be any number that is less than or equal to 11. For instance, t can be 11, 10, 5, 0, -1, or even -100. All of these values satisfy the inequality 13 + t ≤ 24. We can test some values to make sure we get it right. For example, if t = 11, then 13 + 11 = 24, which is “no more than” 24. If t = 10, then 13 + 10 = 23, which is also “no more than” 24.
Visualizing the Solution: Using a Number Line
A number line is a visual tool that really helps to understand inequalities. Let's represent our solution, t ≤ 11, on a number line. First, draw a straight line and mark it with numbers. Put 11 somewhere on the line. Because t can be equal to 11, we'll use a closed circle (filled-in dot) at the number 11. This closed circle indicates that 11 is included in the solution set. Now, because t can be any number less than 11, we shade the number line to the left of 11. This shaded part of the line represents all the possible values of t.
This shows us that any number in the shaded region, all the way to negative infinity, would make our original inequality true. It is very important to remember the difference between the less than or equal to sign (≤) and the less than sign (<). The closed circle on the number line indicates that we include the point on the line; therefore, we use the first sign. The open circle indicates we do not include that point. Using a number line is an awesome way to make inequalities easier to understand, and it helps you visualize the range of solutions. It is always a good idea to draw a number line, as it will always help to give you a complete picture of the possible solutions to an inequality problem. This is useful if you want to make sure your answer is correct. We can see, for example, that the number 10 is a possible solution, because 10 is included on the shaded portion.
Real-World Examples of Inequalities
Inequalities aren't just abstract math concepts; they're all around us. Think about driving: you might see a speed limit sign that says "Maximum speed 65 mph." This is an inequality! You can drive at 65 mph or any speed less than 65 mph, but you can't go more than 65 mph. Another example could be in a budget. Let's say you can spend no more than $50 on groceries. This is an inequality, too. The amount you spend has to be less than or equal to $50. Inequalities help us to describe real-world situations where things are not always equal. They help us set limits, describe constraints, and make decisions based on certain conditions. If you are considering going on a trip and need to know how much you can spend on your trip, or if you are planning a party, then you will have to think about inequalities. Even in everyday situations like these, math plays a vital role. So, next time you see a "no more than" sign or have to stay within a budget, remember the power of inequalities.
More Practice: Try These Problems!
Alright, let's solidify your understanding with some practice problems. Try solving these inequalities on your own:
- The sum of 5 and a number x is no more than 15.
- A number y, increased by 7, is less than 20.
Remember to translate the word problems into inequalities, solve for the variable, and check your answers. You can even draw a number line to visualize your solutions. Don't worry if you get stuck – it's all part of the learning process. Keep practicing, and you'll become an inequality expert in no time!
Conclusion: Mastering Inequalities
Well done, guys! You've now tackled the problem "the sum of 13 and a number t is no more than 24" and have a solid grasp of inequalities. You understand what the phrase "no more than" means, how to translate word problems into mathematical inequalities, and how to solve them. You can visualize the solutions using a number line and have even seen how inequalities pop up in real-world situations. Inequalities are an important skill in math and in life. Keep practicing, exploring, and applying these concepts. You're well on your way to becoming a math whiz! Keep up the awesome work, and happy solving!