Solve Inequalities: Which Number Is The Solution?

Hey guys! Let's break down this math problem together. We're diving into inequalities and figuring out which number from the choices –1, 1, 1.5, and 0 actually works as a solution. This means we need to understand what an inequality is and how to test potential solutions.

Understanding Inequalities

First things first, what exactly is an inequality? Unlike an equation that uses an equals sign (=), an inequality uses symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Think of it this way: instead of saying two things are exactly the same, an inequality shows a range of values that make a statement true. The goal of solving an inequality is to find all the values that satisfy the given condition. This could be a single number, a set of numbers, or even an infinite range of numbers.

Inequalities are super useful in real life. Imagine you're saving up for a new gadget. You might say, "I need to save at least $500." That "at least" is an inequality in action! It means $500 or more will work. In math, we use inequalities to represent situations where we have a range of possible outcomes, not just one specific answer.

When we're given a set of possible solutions, like in this problem, we can test each one individually. It's like a little detective game! We plug each number into the inequality and see if it makes the statement true. If it does, we've found a solution. If not, we move on to the next suspect (I mean, number!). This process of substitution is a fundamental technique in algebra and helps us understand how different values interact with mathematical expressions.

The Question at Hand

Okay, let's get back to the question. We need to know the actual inequality we're dealing with! The prompt gives us the possible answers (–1, 1, 1.5, 0) but doesn't give us the inequality itself. To properly answer this question, we need the inequality. Without it, we're basically trying to solve a puzzle with missing pieces. It's like trying to figure out the end of a story without knowing the beginning!

Let's pretend, for a moment, that the inequality was something like x > 0 (x is greater than 0). Now we have something concrete to work with. We can take each of our answer choices and see if they fit this inequality.

• If x = –1, is –1 > 0? Nope! –1 is a negative number, and negative numbers are less than 0.
• If x = 1, is 1 > 0? Yes! 1 is a positive number and definitely greater than 0.
• If x = 1.5, is 1.5 > 0? You bet! 1.5 is also a positive number and greater than 0.
• If x = 0, is 0 > 0? Nope! 0 is equal to 0, not greater than.

So, if our inequality was x > 0, then 1 and 1.5 would be solutions. See how that works? We plugged in each number and checked if it made the inequality true.

The Importance of the Inequality

The key takeaway here is that we absolutely need the inequality to solve this problem. The answer changes completely depending on what the inequality is. If the inequality was x < 0 (x is less than 0), then only –1 would be a solution from our choices. If it was x ≤ 1 (x is less than or equal to 1), then –1, 0, and 1 would all be solutions.

This highlights a crucial concept in math: context matters! We can't just guess or assume. We need the full problem to find the correct answer. It's like trying to build a house without the blueprints – you might get something that vaguely resembles a house, but it probably won't be very sturdy or functional!

To recap, here's what we've learned:

• Inequalities use symbols like <, >, ≤, and ≥ to show a range of values.
• Solving an inequality means finding the values that make the statement true.
• We can test potential solutions by substituting them into the inequality.
• The actual inequality is essential for solving this type of problem.

So, without the actual inequality, we can't definitively say which answer choice is a solution. But we've practiced the method for solving it once we do have the inequality. Math is all about building those skills step by step!

What if we had a different inequality?

Let’s explore another hypothetical inequality to further solidify our understanding. Suppose the inequality was 2x + 1 ≤ 3. This adds a bit more complexity, but the same principles apply. We still need to test each answer choice to see if it satisfies the inequality.

Why this inequality is interesting:

This inequality involves a variable (x), multiplication, addition, and the “less than or equal to” symbol. It’s a typical example you might encounter in algebra. Solving it requires a few steps, but let's focus on how we'd check if a given number is a solution.

Testing the answer choices:

Now, let's plug in our answer choices (–1, 1, 1.5, 0) into the inequality 2x + 1 ≤ 3 one by one:

• If x = –1: 2(–1) + 1 ≤ 3 becomes –2 + 1 ≤ 3, which simplifies to –1 ≤ 3. Is this true? Yes! –1 is indeed less than or equal to 3. So, –1 is a solution.
• If x = 1: 2(1) + 1 ≤ 3 becomes 2 + 1 ≤ 3, which simplifies to 3 ≤ 3. Is this true? Yes! 3 is equal to 3. Remember, the “≤” symbol means “less than or equal to,” so equality makes the statement true. Therefore, 1 is also a solution.
• If x = 1.5: 2(1.5) + 1 ≤ 3 becomes 3 + 1 ≤ 3, which simplifies to 4 ≤ 3. Is this true? No! 4 is greater than 3. So, 1.5 is not a solution.
• If x = 0: 2(0) + 1 ≤ 3 becomes 0 + 1 ≤ 3, which simplifies to 1 ≤ 3. Is this true? Yes! 1 is less than 3. So, 0 is a solution.

The solutions for 2x + 1 ≤ 3

With this new inequality, we found that –1, 1, and 0 are solutions, while 1.5 is not. This illustrates how the specific inequality drastically affects the answer. Notice how we didn't just blindly plug in numbers; we performed the operations (multiplication and addition) first before comparing the result to 3. This is crucial for correctly evaluating inequalities.

Key Skills Highlighted:

• Substitution: Replacing the variable (x) with a specific number.
• Order of Operations: Following the correct sequence of calculations (PEMDAS/BODMAS).
• Interpreting Inequality Symbols: Understanding the meaning of <, >, ≤, and ≥.
• Logical Reasoning: Determining whether a statement is true or false based on the results.

By working through this example, we've reinforced the importance of having the correct inequality and the process of testing potential solutions. Inequalities, like equations, are fundamental tools in mathematics, and mastering them opens the door to more advanced concepts.

What we’ve Learned and Next Steps

Let's recap what we've covered in our exploration of inequalities and how to determine solutions. We've emphasized the crucial role of the inequality itself in solving these types of problems. Without knowing the specific inequality, we're essentially navigating in the dark. The inequality acts as our guide, telling us the conditions a solution must satisfy.

Core Concepts Revisited:

• The Definition of an Inequality: We clarified that inequalities use symbols like <, >, ≤, and ≥ to express a range of possible solutions, rather than a single, fixed value as in an equation.
• The Substitution Method: We practiced substituting potential solutions into the inequality to check if they make the statement true. This is a cornerstone technique in algebra and problem-solving.
• The Importance of Context: We highlighted that the specific inequality dictates the solution set. The same set of answer choices can yield different solutions depending on the inequality we're working with.
• Real-World Relevance: We touched upon how inequalities show up in everyday scenarios, like saving money or setting limits. This helps connect the abstract math concepts to tangible situations.

Building Upon This Foundation:

Now that we've explored the basics, let's consider what comes next in your mathematical journey. Here are a few key areas to delve into further:

1. Solving Inequalities: We've focused on checking potential solutions. The next step is to learn how to solve inequalities systematically. This involves using algebraic manipulations, similar to solving equations, but with some crucial differences to account for the inequality symbols. For example, multiplying or dividing both sides of an inequality by a negative number requires flipping the inequality sign.
2. Graphing Inequalities: Inequalities can be represented graphically on a number line. This visual representation helps understand the solution set, especially when dealing with ranges of values. Learning to graph inequalities is a valuable skill.
3. Compound Inequalities: These involve two or more inequalities combined with