Solving 2x + 7 > 28 - X: Inequality, Set Notation, Number Line
Let's dive into the world of inequalities! In this comprehensive guide, we'll tackle the inequality 2x + 7 > 28 - x. We'll break it down step-by-step, covering how to isolate x, express the solution using set-builder notation, illustrate it on a number line, and have a thorough discussion about the solution we find. If you've ever felt a bit shaky about inequalities, this is the perfect place to solidify your understanding. So, grab a pen and paper, and let's get started!
A) Isolating x: Finding the Solution
The first crucial step in solving any inequality is to isolate the variable, in this case, x. Our goal is to manipulate the inequality until we have x on one side and a constant on the other. This will give us a clear understanding of the range of values that x can take. So, how do we do this? Let's break down the process with our example inequality, 2x + 7 > 28 - x.
To begin, we want to gather all the x terms on one side of the inequality. A common strategy is to add the term with x from the right side to both sides of the inequality. In our case, we'll add x to both sides. This ensures that we maintain the balance of the inequality, as whatever we do to one side, we must also do to the other. Adding x to both sides of 2x + 7 > 28 - x gives us:
2x + x + 7 > 28 - x + x
Simplifying this, we get:
3x + 7 > 28
Now, we need to isolate the x term further. To do this, we'll eliminate the constant term on the left side, which is +7. We can achieve this by subtracting 7 from both sides of the inequality. Again, maintaining balance is key! Subtracting 7 from both sides gives us:
3x + 7 - 7 > 28 - 7
Which simplifies to:
3x > 21
We're almost there! We now have 3x greater than 21. To completely isolate x, we need to get rid of the coefficient 3. Since 3 is multiplying x, we'll do the inverse operation, which is division. We'll divide both sides of the inequality by 3. As long as we're dividing by a positive number, the direction of the inequality remains the same. Dividing both sides by 3 gives us:
(3x) / 3 > 21 / 3
Which simplifies to:
x > 7
And there we have it! We've successfully isolated x. Our solution is x > 7. This means that any value of x greater than 7 will satisfy the original inequality. Think about it – if x is 8, the inequality holds true. If x is 7.1, it still works. But if x is 7 or less, it won't. This is a fundamental understanding we need before moving on to the next steps.
In summary, isolating x involved a series of algebraic manipulations, ensuring that we kept the inequality balanced throughout the process. We added x to both sides, subtracted 7 from both sides, and finally, divided both sides by 3. Each step was crucial in bringing us closer to our solution, x > 7. Now that we have this solution, let's see how we can express it in set-builder notation.
B) Writing the Solution in Set-Builder Notation
Now that we've successfully isolated x and found that x > 7, the next step is to express this solution in set-builder notation. Set-builder notation is a concise and formal way to describe a set of numbers that satisfy a specific condition. It's a bit like a secret code for mathematicians, allowing them to communicate complex ideas in a clear and standardized way. So, how does it work, and how do we apply it to our solution?
The general form of set-builder notation looks like this:
{ x | condition }
Let's break this down. The curly braces { } indicate that we're talking about a set. The x represents the variable we're interested in – in our case, the variable x from our inequality. The vertical bar | is read as "such that." And the "condition" is the rule or condition that the variable must satisfy to be included in the set. In essence, we're saying, "The set of all x such that... (the condition is true)."
So, how do we translate our solution, x > 7, into this notation? We already have all the pieces! We know our variable is x, and we know the condition is that x must be greater than 7. Putting it all together, the set-builder notation for our solution is:
{ x | x > 7 }
Let's read this aloud to make sure we understand it: "The set of all x such that x is greater than 7." This notation perfectly captures the meaning of our solution. It tells us that we're dealing with a set of numbers, and the only numbers included in that set are those that are greater than 7. It excludes 7 itself, as our inequality is strictly greater than (>) and not greater than or equal to (≥).
Set-builder notation might seem a bit abstract at first, but it's a powerful tool once you get the hang of it. It allows us to express solutions to inequalities and other mathematical problems in a precise and unambiguous way. It's also a stepping stone to more advanced mathematical concepts, so understanding it is crucial. In our case, it beautifully encapsulates the infinite number of values that x can take, as long as they are greater than 7.
Now that we've expressed our solution in set-builder notation, let's visualize it on a number line. This will give us yet another way to understand and interpret our solution.
C) Drawing a Number Line to Depict the Solution
A number line is a fantastic visual tool for representing inequalities and their solutions. It allows us to see the range of values that satisfy an inequality in a clear and intuitive way. Drawing a number line to depict our solution, x > 7, is a simple yet powerful way to solidify our understanding. So, let's grab our imaginary number line and get started!
First, let's draw a horizontal line. This line represents all real numbers, extending infinitely in both directions. We'll mark zero (0) somewhere in the middle as a reference point. Now, we need to locate the critical value in our solution, which is 7. We'll find the approximate position of 7 on the number line and mark it.
Next, we need to indicate that x is greater than 7. This means that all values to the right of 7 on the number line are part of our solution. But how do we visually represent that? Here's where the type of inequality symbol comes into play. Since our solution is x > 7 (strictly greater than), we'll use an open circle at 7. An open circle indicates that 7 itself is not included in the solution set. If our inequality were x ≥ 7 (greater than or equal to), we would use a closed circle to show that 7 is included.
Now, to show that all values greater than 7 are part of the solution, we'll draw an arrow extending from the open circle at 7 to the right, towards positive infinity. This arrow signifies that the solution continues indefinitely in that direction. Any point on the number line that falls under this arrow represents a value of x that satisfies the inequality x > 7.
So, our number line depiction consists of a horizontal line, 0 marked as a reference, an open circle at 7, and an arrow extending to the right from 7. This visual representation clearly shows the solution set: all numbers greater than 7. It excludes 7, as indicated by the open circle, and includes all numbers stretching towards infinity.
The number line provides a valuable complement to our algebraic solution and set-builder notation. It's a visual confirmation of what we've already found, reinforcing our understanding of the inequality and its solution. It also highlights the concept of an infinite solution set – there are infinitely many numbers greater than 7!
Now that we've isolated x, expressed the solution in set-builder notation, and depicted it on a number line, let's move on to a discussion of the solution. This will allow us to explore the implications of our findings and delve deeper into the meaning of the inequality.
D) Discussing the Solution: Implications and Insights
Having solved the inequality 2x + 7 > 28 - x and represented the solution in various ways, it's time to delve into a discussion about what our solution, x > 7, truly means. This is where the real understanding comes in – we're not just crunching numbers, but interpreting their significance in the context of the problem. So, what can we glean from our result?
Firstly, let's reiterate the core concept: x > 7 means that any value of x that is greater than 7 will satisfy the original inequality. This is a crucial understanding. It's not just about finding a single answer; it's about identifying a range of possible values. For example, if we substitute x = 8 into the original inequality, we get:
2(8) + 7 > 28 - 8
16 + 7 > 20
23 > 20
This is true, confirming that 8 is indeed a solution. We could try countless other values greater than 7, and they would all satisfy the inequality. This highlights the infinite nature of the solution set.
Now, let's consider what happens if x is equal to 7. If we substitute x = 7 into the original inequality, we get:
2(7) + 7 > 28 - 7
14 + 7 > 21
21 > 21
This is false! 21 is not greater than 21. This underscores the importance of the "greater than" symbol (>) versus the "greater than or equal to" symbol (≥). Our solution specifically excludes 7, as indicated by the open circle on the number line and the strict inequality symbol in our set-builder notation.
What about values of x less than 7? Let's try x = 6:
2(6) + 7 > 28 - 6
12 + 7 > 22
19 > 22
This is also false. This confirms that values less than 7 do not satisfy the inequality. This is why our number line representation only extends to the right of 7.
The solution x > 7 has practical implications as well. Inequalities are used to model real-world situations where there is a range of acceptable values, rather than a single fixed answer. For example, consider a scenario where you need to earn more than $28, and you already have $7. If you earn $2 per hour, the inequality 2x + 7 > 28 represents the number of hours (x) you need to work. Our solution, x > 10.5, tells us that you need to work more than 10.5 hours to achieve your goal.
In summary, discussing the solution to an inequality involves not just stating the answer, but understanding its implications, verifying its correctness with examples, and relating it to real-world scenarios. The solution x > 7 is not just a mathematical result; it's a statement about the range of values that make the inequality true, and it has practical applications in various contexts. By thoroughly discussing our solution, we solidify our understanding and appreciate the power of inequalities in problem-solving.
Conclusion
In this comprehensive guide, we've taken a deep dive into solving the inequality 2x + 7 > 28 - x. We've covered each step in detail, from isolating x to expressing the solution in set-builder notation and visualizing it on a number line. We've also engaged in a thorough discussion, exploring the implications of our solution and its relevance to real-world scenarios. By working through this example, we've not only honed our algebraic skills but also developed a deeper understanding of inequalities and their applications. Remember, practice makes perfect, so keep tackling those inequalities! For further learning and exploration of inequalities, visit trusted educational resources like Khan Academy's Algebra section.
