Solving Inequality -2(x-5) < 10 With Integer Set

Hey guys! Let's dive into this math problem where we need to solve an inequality with a specific set of integers. It might sound a bit tricky at first, but trust me, we'll break it down step by step so it's super easy to understand. We're given a set of integers from -2 to 2, and our mission is to find out which of these integers satisfy the inequality -2(x-5) < 10. So, grab your thinking caps, and let's get started!

Understanding the Problem

Okay, so the problem gives us a set of numbers: {$x vert -2 leq x leq 2$, where $x$ is an integer }. This simply means we're looking at all the whole numbers between -2 and 2, including -2 and 2 themselves. That gives us the set {-2, -1, 0, 1, 2}. Our main goal here is to solve the inequality: $-2(x-5) < 10$. This means we want to find all the values of $x$ from our set that make this statement true. Inequalities are like equations, but instead of an equals sign, we have symbols like "less than" (<) or "greater than" (>). Solving them involves a bit of algebraic maneuvering, but nothing we can't handle!

Let's break down the key components:

• The Set of Integers: We've got our playground here, the numbers -2, -1, 0, 1, and 2. These are the only numbers we'll consider as possible solutions.
• The Inequality: The heart of the problem is $-2(x-5) < 10$. We need to figure out which numbers from our set, when plugged in for $x$, will make the left side of this inequality less than 10.
• The Goal: Our aim is to identify the subset of integers from our set that satisfies the given inequality. It's like a detective game, where we need to find the suspects that fit the crime (or in this case, the inequality!).

Solving the Inequality

Alright, let's get our hands dirty and solve this inequality! The first step is to simplify the expression $-2(x-5) < 10$. We'll do this by distributing the -2 across the terms inside the parentheses. This means we multiply -2 by both $x$ and -5.

Step-by-Step Solution

1. Distribute -2:

   $-2 * x = -2x$

   $-2 * -5 = 10$

   So, the inequality becomes $-2x + 10 < 10$.

2. Isolate the term with x:

   We want to get the term with $x$ by itself on one side of the inequality. To do this, we subtract 10 from both sides:

   $-2x + 10 - 10 < 10 - 10$

   This simplifies to $-2x < 0$.

3. Solve for x:

   Now, we need to get $x$ all alone. Since $x$ is being multiplied by -2, we'll divide both sides of the inequality by -2. But here's a crucial rule to remember: when you divide or multiply both sides of an inequality by a negative number, you must flip the direction of the inequality sign.

   So, $-2x < 0$ becomes:

   $x > 0$ (Notice how the "less than" sign became a "greater than" sign!)

What Does x > 0 Mean?

This result, $x > 0$, tells us that $x$ must be greater than 0. In other words, any number larger than 0 will satisfy this inequality. But remember, we're not dealing with all numbers here – we have a specific set of integers to consider.

Finding the Solution Within the Set

Okay, we've solved the inequality and found that $x > 0$. Now, let's bring back our set of integers: {-2, -1, 0, 1, 2}. We need to figure out which numbers from this set are greater than 0. This is where it gets pretty straightforward!

Identifying the Integers Greater Than 0

Looking at our set, the integers that are greater than 0 are:

• 1
• 2

That's it! These are the only two numbers from our set that fit the condition $x > 0$.

The Solution Set

So, the solution to the inequality $-2(x-5) < 10$ within the given set of integers is {1, 2}. These are the only values of $x$ from our set that make the inequality true. We've successfully navigated the algebraic steps and pinpointed the exact solutions. Nice work, guys!

Verifying the Solution

To be absolutely sure we've got the correct answer, it's always a good idea to verify our solution. This means we'll plug our solution set, {1, 2}, back into the original inequality, $-2(x-5) < 10$, to see if it holds true.

Testing x = 1

Let's substitute $x$ with 1:

$-2(1 - 5) < 10$

$-2(-4) < 10$

$8 < 10$

This is true! 8 is indeed less than 10, so $x = 1$ is a valid solution.

Testing x = 2

Now, let's substitute $x$ with 2:

$-2(2 - 5) < 10$

$-2(-3) < 10$

$6 < 10$

This is also true! 6 is less than 10, confirming that $x = 2$ is also a valid solution.

Why Other Integers Don't Work

Just for completeness, let's quickly see why the other integers in our set (-2, -1, and 0) don't work:

• x = -2: $-2(-2 - 5) = -2(-7) = 14$, which is not less than 10.
• x = -1: $-2(-1 - 5) = -2(-6) = 12$, which is not less than 10.
• x = 0: $-2(0 - 5) = -2(-5) = 10$, which is not less than 10 (it's equal to 10, but not less).

So, we've thoroughly verified that our solution set {1, 2} is correct. We plugged in the values, and they satisfied the original inequality. This step is like the final stamp of approval on our work! It gives us confidence that we've nailed the problem.

Conclusion

Alright, awesome job, everyone! We've successfully tackled this problem from start to finish. We started with a set of integers and an inequality, and through careful algebraic manipulation and verification, we found the solution set. Remember, the key to solving these types of problems is to:

• Understand the Problem: Know what you're given and what you're trying to find.
• Solve the Inequality: Use algebraic techniques to isolate the variable.
• Consider the Set: Identify the solutions that fit within the given set.
• Verify the Solution: Plug the solutions back into the original inequality to make sure they work.

Math can sometimes feel like a puzzle, but with a systematic approach and a bit of practice, you can solve anything! Keep up the great work, and I'll catch you in the next math adventure!