Solving Inequalities: A Step-by-Step Guide
Hey guys! Today, we're diving into the world of inequalities and tackling the problem: solve the following inequalities in β:
a) x / 3 - 1 β₯ 1
b) 5(x - 1) - 3 < 3x + 2
Don't worry, it's not as scary as it looks! We'll break it down step-by-step, so you can confidently solve these and other similar problems. Inequalities are a fundamental concept in mathematics, appearing everywhere from basic algebra to advanced calculus. Mastering them is crucial for any aspiring mathematician, engineer, or scientist. So, let's get started and unlock the secrets of inequalities!
Understanding Inequalities
Before we jump into solving, let's quickly recap what inequalities are. Unlike equations that state two expressions are equal, inequalities compare expressions using symbols like:
- > (greater than)
- < (less than)
- β₯ (greater than or equal to)
- β€ (less than or equal to)
Think of them as a way to express a range of possible solutions rather than a single value. When we talk about solving inequalities in β, we mean finding all the real numbers that satisfy the given condition. Remember, real numbers encompass pretty much any number you can think of β integers, fractions, decimals, even irrational numbers like β2 or Ο. Understanding this is key to accurately interpreting our solutions later on.
Solving inequalities involves similar techniques to solving equations, but with a crucial difference: multiplying or dividing by a negative number flips the inequality sign. Keep this in mind, as itβs a common pitfall that can lead to incorrect answers. For example, if we have -x > 2, dividing both sides by -1 gives us x < -2, not x > -2.
Solving Inequality a) x / 3 - 1 β₯ 1
Let's start with the first inequality:
a) x / 3 - 1 β₯ 1
Our goal is to isolate x on one side of the inequality. Think of it like peeling away layers to reveal the x at the core.
Step 1: Add 1 to both sides
To get rid of the -1 on the left side, we add 1 to both sides of the inequality. Remember, whatever we do to one side, we must do to the other to maintain the balance.
x / 3 - 1 + 1 β₯ 1 + 1
This simplifies to:
x / 3 β₯ 2
Step 2: Multiply both sides by 3
Now, we want to get rid of the division by 3. So, we multiply both sides by 3:
3 * (x / 3) β₯ 3 * 2
This simplifies to:
x β₯ 6
Solution
So, the solution to the inequality x / 3 - 1 β₯ 1 is x β₯ 6. This means any real number greater than or equal to 6 will satisfy the inequality. We can represent this solution graphically on a number line, with a closed circle at 6 (since 6 is included) and an arrow extending to the right, indicating all numbers greater than 6.
It's always a good idea to check your solution. Pick a number greater than or equal to 6, say 9. Substitute it back into the original inequality: 9 / 3 - 1 β₯ 1, which simplifies to 3 - 1 β₯ 1, or 2 β₯ 1. This is true, so our solution seems correct. Conversely, if we pick a number less than 6, say 3, we get 3 / 3 - 1 β₯ 1, which simplifies to 0 β₯ 1, which is false. This further confirms our solution.
Solving Inequality b) 5(x - 1) - 3 < 3x + 2
Now, let's tackle the second inequality:
b) 5(x - 1) - 3 < 3x + 2
This one looks a bit more complex, but we'll use the same principles to solve it. The key here is to carefully apply the order of operations and distribute properly.
Step 1: Distribute the 5
First, we need to distribute the 5 across the parentheses:
5 * x - 5 * 1 - 3 < 3x + 2
This simplifies to:
5x - 5 - 3 < 3x + 2
Step 2: Combine like terms
Next, let's combine the constant terms on the left side:
5x - 8 < 3x + 2
Step 3: Subtract 3x from both sides
Now, we want to get all the x terms on one side. Let's subtract 3x from both sides:
5x - 8 - 3x < 3x + 2 - 3x
This simplifies to:
2x - 8 < 2
Step 4: Add 8 to both sides
Next, we isolate the x term by adding 8 to both sides:
2x - 8 + 8 < 2 + 8
This simplifies to:
2x < 10
Step 5: Divide both sides by 2
Finally, we divide both sides by 2 to solve for x:
2x / 2 < 10 / 2
This simplifies to:
x < 5
Solution
The solution to the inequality 5(x - 1) - 3 < 3x + 2 is x < 5. This means any real number less than 5 will satisfy this inequality. Again, we can visualize this on a number line with an open circle at 5 (since 5 is not included) and an arrow extending to the left.
Let's check our solution. If we pick a number less than 5, say 0, and substitute it into the original inequality, we get 5(0 - 1) - 3 < 3(0) + 2, which simplifies to -8 < 2. This is true. If we pick a number greater than or equal to 5, say 5, we get 5(5 - 1) - 3 < 3(5) + 2, which simplifies to 17 < 17, which is false. Our solution checks out!
Key Takeaways for Solving Inequalities
Solving inequalities might seem tricky at first, but with practice, you'll get the hang of it. Here's a quick recap of the key steps:
- Simplify: Distribute, combine like terms, and generally tidy up the inequality.
- Isolate the variable: Use addition, subtraction, multiplication, and division to get the variable on one side of the inequality.
- Remember the flip: If you multiply or divide by a negative number, flip the inequality sign!
- Check your solution: Substitute a value from your solution back into the original inequality to ensure it holds true.
- Visualize the solution: Use a number line to represent the solution set graphically. This helps in understanding the range of values that satisfy the inequality.
Inequalities in the Real World
You might be wondering, βWhen will I ever use this?β Inequalities are incredibly useful in many real-world situations. They help us model scenarios where we're dealing with ranges of values rather than exact numbers. For instance:
- Budgeting: You might have a budget constraint like βI can spend no more than $100 on groceries.β This translates to an inequality: spending β€ $100.
- Speed limits: A speed limit is an inequality: speed β€ maximum speed.
- Temperature ranges: You might need to keep a chemical reaction within a certain temperature range, represented by an inequality.
- Optimization problems: In business and engineering, inequalities are used to find the best possible solution within certain constraints.
Practice Makes Perfect
The best way to master inequalities is to practice! Work through various examples, and don't be afraid to make mistakes. Each mistake is a learning opportunity. Try solving similar problems with different coefficients and constants. Challenge yourself with more complex inequalities involving multiple steps or absolute values. Online resources, textbooks, and your math teacher are all great sources of practice problems.
Conclusion
So, there you have it! We've successfully solved the inequalities x / 3 - 1 β₯ 1 and 5(x - 1) - 3 < 3x + 2. Remember, the key is to break down the problem into smaller steps, carefully apply the rules, and always check your solution. Keep practicing, and you'll become an inequality-solving pro in no time! Understanding inequalities is a crucial skill in mathematics, with applications far beyond the classroom. By mastering these concepts, you're building a solid foundation for future mathematical endeavors and real-world problem-solving. Keep up the great work, guys!