Solving Compound Inequalities & Graphing Solutions

Hey guys! Today, we're diving into the world of compound inequalities. Don't worry, it sounds more intimidating than it actually is. Essentially, we're going to solve two inequalities and then see where their solutions overlap or combine. We'll tackle the specific example: solving $2x + 5 > 3$ and $4x + 7 < 15$, then we will graph the solution. Let's jump right in!

Understanding Compound Inequalities

Before we get our hands dirty with the actual problem, let's quickly recap what compound inequalities are all about. Compound inequalities are basically two inequalities joined together by either an "and" or an "or." The "and" means that both inequalities have to be true at the same time. The "or" means that at least one of the inequalities has to be true. In our case, we have an "and" compound inequality, so we need to find the values of x that satisfy both $2x + 5 > 3$ and $4x + 7 < 15$.

When dealing with compound inequalities, it's super important to pay close attention to the connective word ("and" or "or"). This word dictates how we interpret and combine the individual solutions. For "and" inequalities, we're looking for the intersection of the solution sets – the values that make both inequalities true. For "or" inequalities, we're looking for the union of the solution sets – the values that make at least one of the inequalities true. Getting this distinction clear from the start will save you a lot of headaches down the road. Remember, it's all about carefully reading the problem and understanding what it's asking you to find.

Also, remember the basic rules of solving inequalities. You can add or subtract the same number from both sides without changing the inequality. You can multiply or divide both sides by a positive number without changing the inequality. But, and this is a big but, if you multiply or divide by a negative number, you have to flip the inequality sign! Keep these rules in the back of your mind as we move forward. With these concepts in mind, we're ready to solve the system.

Solving the First Inequality: $2x + 5 > 3$

Okay, let's take the first inequality, which is $2x + 5 > 3$. Our mission here is to isolate x on one side of the inequality. To do that, we'll start by subtracting 5 from both sides. This gives us:

$2x + 5 - 5 > 3 - 5$

Simplifying, we get:

$2x > -2$

Now, to get x all by itself, we need to divide both sides by 2:

$2x / 2 > -2 / 2$

Which simplifies to:

$x > -1$

So, the solution to the first inequality is x is greater than -1. That means any number bigger than -1 will make the inequality $2x + 5 > 3$ true. For example, if we plug in x = 0, we get $2(0) + 5 > 3$, which simplifies to $5 > 3$, and that's definitely true!

Make sure to keep track of this result, as we'll need it later when we combine it with the solution of the second inequality. Always double-check your work to make sure you haven't made any arithmetic errors along the way. A small mistake in one step can throw off the entire solution. We want to ensure our results are as accurate as possible!

Solving the Second Inequality: $4x + 7 < 15$

Now, let's tackle the second inequality: $4x + 7 < 15$. Just like before, our goal is to get x by itself on one side. To start, we'll subtract 7 from both sides:

$4x + 7 - 7 < 15 - 7$

Simplifying this, we have:

$4x < 8$

Next, we divide both sides by 4 to isolate x:

$4x / 4 < 8 / 4$

Which gives us:

$x < 2$

So, the solution to the second inequality is x is less than 2. This means any number smaller than 2 will satisfy the inequality $4x + 7 < 15$. If we try x = 1, we get $4(1) + 7 < 15$, which simplifies to $11 < 15$, and that's also true!

Just like with the first inequality, make sure you carefully review each step to ensure accuracy. It's easy to make a small mistake with the arithmetic, especially when dealing with multiple steps. And as before, keep track of this solution, as we will now move on to graphing the solutions.

Combining the Solutions and Graphing

Alright, we've solved both inequalities! We found that $x > -1$ and $x < 2$. Now, we need to combine these solutions since our original problem had an "and" condition. This means we're looking for the values of x that satisfy both inequalities simultaneously.

In other words, x has to be greater than -1 and less than 2. We can write this as a single compound inequality: $-1 < x < 2$. This tells us that x lies between -1 and 2, but it doesn't include -1 or 2 themselves.

Now, let's graph this solution. We'll draw a number line. Place open circles at -1 and 2 because x cannot equal -1 or 2. Then, we'll shade the region between -1 and 2 to show all the values of x that satisfy the compound inequality. The graph should clearly show an open interval between -1 and 2, indicating that the solution set includes all numbers between these two values but not the values themselves.

Key points for Graphing:

• Open Circle: Represents that the endpoint is not included in the solution (used for > and <).
• Closed Circle: Represents that the endpoint is included in the solution (used for ≥ and ≤).
• Shading: Indicates the range of values that satisfy the inequality.

When graphing compound inequalities, it's always a good idea to double-check that your graph accurately reflects the solution you found algebraically. Make sure the endpoints are represented correctly (open or closed circles) and that the correct region is shaded. A clear and accurate graph is a great way to communicate your solution effectively.

Final Answer

So, to wrap it all up, we solved the compound inequality $2x + 5 > 3$ and $4x + 7 < 15$. We found that the solution is $-1 < x < 2$. This means that x can be any number between -1 and 2, not including -1 and 2. We also graphed this solution on a number line, using open circles at -1 and 2 and shading the region in between.

Remember, solving compound inequalities involves breaking them down into individual inequalities, solving each one separately, and then combining the solutions based on whether the inequalities are joined by "and" or "or". Graphing the solution provides a visual representation of the range of values that satisfy the compound inequality.

And that's it! You've successfully solved and graphed a compound inequality. Keep practicing, and you'll become a pro in no time. Great job, everyone!