Solve Compound Inequalities: A Step-by-Step Guide

Hey everyone! Today, we're diving into the world of compound inequalities. These might seem a bit tricky at first, but trust me, once you get the hang of them, they're actually pretty straightforward. We'll be tackling two specific examples in this guide, breaking down each step so you can confidently solve these types of problems on your own. So, grab your pencils and notebooks, and let's get started!

Understanding Compound Inequalities

Before we jump into the examples, let's quickly define what compound inequalities are. Basically, they're two or more inequalities combined into one statement. These inequalities are usually joined by the words "and" or "or." The word "and" means that both inequalities must be true at the same time, while "or" means that at least one of the inequalities must be true. Knowing this distinction is crucial for finding the correct solution set.

To truly understand compound inequalities, it's helpful to visualize them on a number line. When dealing with "and" inequalities, we're looking for the intersection of the solutions – the values that satisfy both inequalities. On the number line, this is represented by the overlapping region. On the other hand, "or" inequalities are about the union of the solutions. We include all values that satisfy either inequality, which means we're looking at the combined regions on the number line.

Think of it this way: if the problem says “and”, it’s like saying you need to meet two specific requirements simultaneously – both conditions must be true. But if it’s “or”, it's more flexible – you only need to meet one of the requirements. This fundamental difference in how "and" and "or" work dictates how we solve and interpret these inequalities. We'll see this in action as we work through our examples, so keep this in mind.

Example 1: Solving 19. $5-2x > 17$ or $3-4x < 19$

Let's kick things off with our first example: $5-2x > 17$ or $3-4x < 19$. This is an "or" compound inequality, which means we'll be solving each inequality separately and then combining their solutions. Remember, at least one of the inequalities needs to be true for a value to be part of the solution.

Step 1: Solve the First Inequality ($5-2x > 17$)

First, we'll focus on $5-2x > 17$. Our goal here is to isolate $x$ on one side of the inequality. To do this, we'll start by subtracting 5 from both sides:

$5 - 2x - 5 > 17 - 5$

This simplifies to:

$-2x > 12$

Now, we need to divide both sides by -2 to get $x$ by itself. But, and this is very important, when we divide or multiply an inequality by a negative number, we need to flip the inequality sign. So, we get:

$x < -6$

So, the solution to the first inequality is $x$ is less than -6. This means any number smaller than -6 will satisfy this part of our compound inequality.

Step 2: Solve the Second Inequality ($3-4x < 19$)

Now let's tackle the second inequality: $3-4x < 19$. Just like before, we want to isolate $x$. We'll start by subtracting 3 from both sides:

$3 - 4x - 3 < 19 - 3$

This simplifies to:

$-4x < 16$

Again, we need to divide both sides by a negative number (-4), so we flip the inequality sign:

$x > -4$

So, the solution to the second inequality is $x$ is greater than -4. This means any number bigger than -4 will satisfy this part of our compound inequality.

Step 3: Combine the Solutions

Now for the crucial part: combining the solutions. Remember, this is an "or" inequality, so we're looking for all values that satisfy either $x < -6$ or $x > -4$. To visualize this, imagine a number line. We have an open circle at -6 (because $x$ is strictly less than -6, not less than or equal to) and shade everything to the left. We also have an open circle at -4 and shade everything to the right.

The combined solution is all real numbers less than -6 or greater than -4. In interval notation, we can write this as $(-\infty, -6) \cup (-4, \infty)$. The $\cup$ symbol means "union," indicating we're combining the two intervals.

In simple terms, if you pick any number smaller than -6, it works. If you pick any number bigger than -4, it works. The numbers between -6 and -4 don't work. That's the solution to this compound inequality!

Example 2: Solving 22. $-4x < 12$ or $3x < -21$

Alright, let's move on to our second example: $-4x < 12$ or $3x < -21$. This is another "or" compound inequality, so we'll follow the same steps as before: solve each inequality separately and then combine the results.

Step 1: Solve the First Inequality ($-4x < 12$)

We'll start with $-4x < 12$. To isolate $x$, we need to divide both sides by -4. Don't forget to flip the inequality sign when dividing by a negative number!

$x > -3$

So, the solution to the first inequality is $x$ is greater than -3. Any number larger than -3 will satisfy this inequality.

Step 2: Solve the Second Inequality ($3x < -21$)

Next up is $3x < -21$. This one's a bit simpler because we're dividing by a positive number. We divide both sides by 3:

$x < -7$

So, the solution to the second inequality is $x$ is less than -7. Any number smaller than -7 will satisfy this inequality.

Step 3: Combine the Solutions

Time to put it all together. We have an "or" inequality, so we want values that satisfy either $x > -3$ or $x < -7$. Let's visualize this on a number line again. We have an open circle at -3 and shade everything to the right. We also have an open circle at -7 and shade everything to the left.

The combined solution is all real numbers less than -7 or greater than -3. In interval notation, this is $(-\infty, -7) \cup (-3, \infty)$.

Essentially, any number smaller than -7 works, and any number bigger than -3 works. The numbers in between -7 and -3 don't. You're getting the hang of this, right?

Key Takeaways for Solving Compound Inequalities

Before we wrap things up, let's quickly recap the key steps to solving compound inequalities. These tips will help you tackle any similar problems you encounter:

1. Identify the type: Is it an "and" or an "or" inequality? This determines how you'll combine the solutions.
2. Solve each inequality separately: Isolate the variable in each inequality using the same techniques you'd use for regular equations.
3. Remember to flip the inequality sign: When you multiply or divide by a negative number, flip the inequality sign.
4. Combine the solutions:
   • For "and" inequalities, find the intersection (the overlapping region on the number line).
   • For "or" inequalities, find the union (all regions on the number line).
5. Express the solution: You can write the solution as an inequality, on a number line, or in interval notation. Choose the method that best suits the problem and your understanding.

Practice Makes Perfect

The best way to master solving compound inequalities is to practice! Work through more examples, try different variations, and don't be afraid to make mistakes. Each mistake is a learning opportunity. The more you practice, the more comfortable and confident you'll become with these types of problems.

So there you have it, guys! A comprehensive guide to solving compound inequalities with two detailed examples. We've covered everything from the basic definitions to the specific steps you need to take. Now it's your turn to put your knowledge to the test. Happy solving!