Hey guys! Today, we're diving into the world of inequalities and how to solve them. Specifically, we're going to tackle the inequality -3p + 5 ≤ 20. Not only will we find the solution, but we'll also learn how to represent it graphically. So, let's get started!
Understanding Inequalities
Before we jump into solving, let's quickly recap what inequalities are. Unlike equations that show equality (=), inequalities show relationships where one side is not necessarily equal to the other. We use symbols like:
<(less than)>(greater than)≤(less than or equal to)≥(greater than or equal to)
Think of it like this: if you're saying x < 5, you mean x can be any number smaller than 5, but not 5 itself. If you say x ≤ 5, then x can be any number smaller than or equal to 5, including 5. This subtle difference is crucial when we graph our solutions.
Why Inequalities Matter
You might be wondering, why bother with inequalities? Well, in the real world, things aren't always exact. Think about speed limits on a road (speed ≤ 65 mph), budget constraints (spending ≤ $100), or the minimum score you need to pass a test (score ≥ 70). Inequalities help us model these situations where there's a range of acceptable values, not just one specific number. Understanding this concept is key to applying math in practical scenarios.
Key Concepts in Solving Inequalities
Solving inequalities is very similar to solving equations, but there's one major twist. Most of the time, you can add, subtract, multiply, or divide both sides of the inequality by the same number, just like with equations. However, there's a critical exception: when you multiply or divide both sides by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number changes the direction of the inequality. For instance, if 2 < 4, multiplying both sides by -1 gives -2 > -4. See how the < flipped to >? Keep this in mind as we solve our problem.
Now that we've laid the groundwork, let's solve the inequality -3p + 5 ≤ 20 step by step.
Step-by-Step Solution of -3p + 5 ≤ 20
Okay, let's get our hands dirty with the actual math! Our goal is to isolate p on one side of the inequality. We'll do this using the same algebraic principles we use for equations, with that one important rule about flipping the sign when multiplying or dividing by a negative number.
Step 1: Isolate the Term with 'p'
Our inequality is -3p + 5 ≤ 20. The first thing we want to do is get the term with p (-3p) by itself on one side. To do this, we'll subtract 5 from both sides of the inequality. Remember, whatever we do to one side, we must do to the other to maintain the balance.
-3p + 5 - 5 ≤ 20 - 5
This simplifies to:
-3p ≤ 15
Great! We've successfully isolated the term with p. Now, we need to get p completely alone.
Step 2: Isolate 'p'
Currently, p is being multiplied by -3. To undo this multiplication, we need to divide both sides of the inequality by -3. This is where our special rule comes into play! Since we're dividing by a negative number, we must flip the inequality sign.
-3p / -3 ≥ 15 / -3
Notice that the ≤ has changed to ≥. This is absolutely crucial. If you forget to flip the sign, you'll end up with the wrong solution.
Now, let's simplify:
p ≥ -5
We've done it! We've solved the inequality. Our solution is p ≥ -5. This means p can be any number greater than or equal to -5.
Step 3: Expressing the Solution in Interval Notation
Interval notation is a concise way to represent a range of numbers. It uses brackets and parentheses to indicate whether the endpoints are included in the solution.
[and](square brackets) mean the endpoint is included (less than or equal to, or greater than or equal to).(and)(parentheses) mean the endpoint is not included (less than, or greater than).∞(infinity) and-∞(negative infinity) always use parentheses because infinity is not a specific number, it's a concept.
So, how do we express p ≥ -5 in interval notation? Well, p can be -5, or any number bigger than -5, all the way up to infinity. This is written as:
[-5, ∞)
The square bracket [ next to -5 means -5 is included, and the parenthesis ) next to infinity means infinity is not included (which makes sense, because you can't actually reach infinity!).
Now that we have our solution in interval notation, let's visualize it on a graph.
Graphing the Solution
Graphing the solution helps us see the range of values that satisfy the inequality. We'll use a number line to represent all possible values of p and then highlight the part that represents our solution, p ≥ -5.
Step 1: Draw a Number Line
Draw a horizontal line and mark some numbers on it. Make sure to include -5, as that's our key point. You can include other integers around -5, like -6, -4, -3, etc., to give context.
Step 2: Mark the Endpoint
We need to mark -5 on our number line. But how do we indicate that -5 is included in the solution (due to the ≥)? We use a closed circle or a filled-in dot. This visually tells us that -5 is part of the solution set.
If the inequality were p > -5 (strictly greater than), we'd use an open circle at -5 to show that -5 is not included.
Step 3: Shade the Solution Region
Our solution is p ≥ -5, which means p can be -5 or any number greater than -5. On the number line, numbers greater than -5 are to the right of -5. So, we'll shade the number line to the right of our closed circle at -5. This shaded region represents all the values of p that satisfy the inequality.
To further emphasize that the solution continues indefinitely, we usually draw an arrow at the end of the shaded region, pointing towards infinity.
Putting It All Together
So, our graph will have a number line, a filled-in dot at -5, and a shaded region extending to the right, with an arrow indicating it continues to infinity. This visual representation perfectly captures the meaning of p ≥ -5.
Common Mistakes to Avoid
Alright, we've solved the inequality and graphed the solution. But before we wrap up, let's talk about some common pitfalls students often encounter. Avoiding these mistakes will help you ace your inequality problems!
Forgetting to Flip the Sign
This is, without a doubt, the most common mistake. Remember, whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. If you forget this, your solution will be completely wrong. Double-check this step every time you solve an inequality involving multiplication or division by a negative number.
Confusing Open and Closed Circles
On the graph, an open circle means the endpoint is not included in the solution ( < or > ), while a closed circle means it is included ( ≤ or ≥ ). Mixing these up will lead to an incorrect visual representation of your solution. Always pay close attention to the inequality symbol when deciding whether to use an open or closed circle.
Misinterpreting Interval Notation
Interval notation is a useful shorthand, but it can be confusing if you don't understand the conventions. Remember that square brackets [ and ] mean the endpoint is included, while parentheses ( and ) mean it's not. Also, infinity (∞) and negative infinity (-∞) always get parentheses. Practice converting between inequalities and interval notation to solidify your understanding.
Not Checking Your Solution
It's always a good idea to check your solution, especially on a test. Pick a number within your solution set and plug it back into the original inequality. If the inequality holds true, your solution is likely correct. For example, in our problem p ≥ -5, we could pick p = 0 (which is greater than -5) and plug it into -3p + 5 ≤ 20. We get -3(0) + 5 ≤ 20, which simplifies to 5 ≤ 20, which is true! This gives us confidence that our solution is correct.
By being aware of these common mistakes, you can significantly improve your accuracy when solving and graphing inequalities.
Conclusion
Woohoo! We've successfully solved the inequality -3p + 5 ≤ 20, found the solution in interval notation ([-5, ∞)), and graphed it on a number line. We also discussed common mistakes to watch out for. You're now well-equipped to tackle similar inequality problems. Remember the key concepts, especially that flipping-the-sign rule, and practice, practice, practice! You've got this!