Solving Compound Inequalities: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of compound inequalities. Specifically, we're going to tackle one that looks a little intimidating at first glance: $-24 ≤ -3x - 3$ or $-39 > -3x - 3$. Don't worry, we'll break it down step by step, so you'll be a pro in no time! We'll not only solve for x but also graph the solution on a number line. So, grab your pencils and let's get started!

Understanding Compound Inequalities

First, let's understand what we're dealing with. A compound inequality is essentially two inequalities joined by either an "or" or an "and." The "or" means that the solution includes all values that satisfy either inequality. The "and" means the solution includes only values that satisfy both inequalities simultaneously. In our case, we have an "or," which means we need to find all the x values that make either $-24 ≤ -3x - 3$ true OR $-39 > -3x - 3$ true. Got it? Awesome!

When approaching compound inequalities, it's often easiest to solve each inequality separately and then combine the solutions based on whether it's an "or" or an "and." This keeps things organized and lessens the chance of making mistakes. Think of it like tackling two smaller problems instead of one big, scary one.

Also, remember that when you multiply or divide both sides of an inequality by a negative number, you need to flip the direction of the inequality sign. This is a crucial rule that students often forget, so keep it in mind! It's like the inequality sign is saying, "Hey, we're switching things up!" For instance, if you have $-x < 5$, multiplying both sides by -1 gives you $x > -5$.

Lastly, always simplify each inequality as much as possible before combining the solutions. This means getting the x term by itself on one side and everything else on the other. Simplifying early makes the subsequent steps much easier and cleaner. Alright, enough talk, let’s get to the actual solving!

Solving the First Inequality: $-24 ≤ -3x - 3$

Okay, let's kick things off by solving the first inequality: $-24 ≤ -3x - 3$. Our goal here is to isolate x on one side of the inequality. To do that, we'll start by adding 3 to both sides. This gets rid of the -3 that's hanging out with the x term. So, we have:

$-24 + 3 ≤ -3x - 3 + 3$

This simplifies to:

$-21 ≤ -3x$

Now, we need to get x all by itself. To do this, we'll divide both sides by -3. But remember that crucial rule: when we divide by a negative number, we need to flip the inequality sign! So, we get:

$\frac{-21}{-3} ≥ \frac{-3x}{-3}$

Simplifying this gives us:

$7 ≥ x$

Which we can also write as:

$x ≤ 7$

So, the solution to our first inequality is $x ≤ 7$. This means that any value of x that is less than or equal to 7 will satisfy the inequality $-24 ≤ -3x - 3$. Make sense? Great! Now, let's move on to the second inequality.

Solving the Second Inequality: $-39 > -3x - 3$

Alright, let's tackle the second inequality: $-39 > -3x - 3$. Just like before, our mission is to isolate x. We'll start by adding 3 to both sides of the inequality:

$-39 + 3 > -3x - 3 + 3$

This simplifies to:

$-36 > -3x$

Now, we need to divide both sides by -3 to get x by itself. And yes, you guessed it, we need to flip the inequality sign again because we're dividing by a negative number. So, we have:

$\frac{-36}{-3} < \frac{-3x}{-3}$

Simplifying this gives us:

$12 < x$

Which we can also write as:

$x > 12$

So, the solution to our second inequality is $x > 12$. This means that any value of x that is greater than 12 will satisfy the inequality $-39 > -3x - 3$.

Combining the Solutions

Now that we've solved both inequalities, we need to combine their solutions. Remember that our original problem was a compound inequality with an "or." This means we need to find all values of x that satisfy either $x ≤ 7$ OR $x > 12$. There's no overlap between these two conditions; a number can't be both less than or equal to 7 AND greater than 12 at the same time. Therefore, our combined solution is simply the union of the two individual solutions.

So, the final solution to the compound inequality $-24 ≤ -3x - 3$ or $-39 > -3x - 3$ is:

$x ≤ 7$ or $x > 12$

This means that x can be any number less than or equal to 7, or it can be any number greater than 12. Numbers between 7 and 12 (including 8, 9, 10, 11, and 12) are not part of the solution.

Graphing the Solution on a Number Line

Finally, let's graph our solution on a number line. This will give us a visual representation of all the values of x that satisfy the compound inequality.

1. Draw a number line: Draw a straight line and mark some numbers on it, including 7 and 12. Make sure to include numbers smaller than 7 and larger than 12 as well.
2. Mark $x ≤ 7$: Since x can be equal to 7, we'll use a closed circle (or a filled-in dot) on the number 7. Then, we'll draw a line extending to the left from 7, indicating that all numbers less than 7 are also part of the solution. Shade this line to make it clear.
3. Mark $x > 12$: Since x cannot be equal to 12 (it has to be strictly greater than 12), we'll use an open circle (or an empty dot) on the number 12. Then, we'll draw a line extending to the right from 12, indicating that all numbers greater than 12 are also part of the solution. Shade this line as well.

The graph will show two distinct shaded regions: one extending to the left from 7 (including 7) and one extending to the right from 12 (excluding 12). The space between 7 and 12 will be unshaded, indicating that those numbers are not part of the solution.

Conclusion

And there you have it! We've successfully solved the compound inequality $-24 ≤ -3x - 3$ or $-39 > -3x - 3$ and graphed the solution on a number line. Remember the key steps: solve each inequality separately, pay attention to flipping the inequality sign when dividing by a negative number, and combine the solutions based on the "or" or "and" condition. Keep practicing, and you'll become a master of compound inequalities in no time!