Solving The Inequality: (5 - 3x)/2 >= -x - 1
Hey everyone! Today, we're diving into the exciting world of inequalities, specifically tackling the problem (5 - 3x)/2 >= -x - 1. Inequalities might seem a bit intimidating at first, but trust me, once you grasp the fundamental concepts, they become quite manageable. We'll break down each step, explain the reasoning behind it, and make sure you're comfortable with the entire process. So, buckle up and let's get started!
Understanding Inequalities
Before we jump into the solution, let's quickly recap what inequalities are all about. Unlike equations, which aim to find specific values that make the expressions on both sides equal, inequalities deal with relationships where one side is greater than, less than, greater than or equal to, or less than or equal to the other side. The symbols we use are: > (greater than), < (less than), >= (greater than or equal to), and <= (less than or equal to). Understanding these symbols is crucial for interpreting and solving inequalities correctly. Inequalities are used everywhere, from determining the range of acceptable values in engineering to modeling economic scenarios.
When solving inequalities, our goal is to isolate the variable (in this case, 'x') on one side, just like solving equations. However, there's one key difference: multiplying or dividing both sides by a negative number flips the direction of the inequality sign. This is a critical rule to remember, as it's a common source of errors. To really understand this, think of a number line. Multiplying by a negative number essentially reflects the numbers across zero, changing their order. Let’s look at a simple example: If 2 < 3, then multiplying both sides by -1 gives us -2 > -3. See how the inequality sign flipped? Keep this in mind as we solve our inequality!
Step-by-Step Solution
Let's solve this inequality step by step. Each step will be explained in detail so you can understand the logic behind each manipulation.
1. Clear the Fraction
The first step in solving (5 - 3x)/2 >= -x - 1 is to eliminate the fraction. Fractions can make things look complicated, but they're easily dealt with by multiplying both sides of the inequality by the denominator. In this case, our denominator is 2. So, we multiply both sides by 2:
2 * [(5 - 3x)/2] >= 2 * (-x - 1)
This simplifies to:
5 - 3x >= -2x - 2
Multiplying both sides by the same positive number doesn't change the direction of the inequality, which is why we can proceed without flipping the sign. We've now transformed the inequality into a more manageable form, free from fractions. This step is crucial because it simplifies the equation, making it easier to work with. Remember, the goal is to isolate 'x', and clearing the fraction is a significant move towards achieving that.
2. Group the 'x' Terms
Now, let's gather all the terms containing 'x' on one side of the inequality. This is similar to solving equations, where we want to keep like terms together. To do this, we'll add 3x to both sides:
5 - 3x + 3x >= -2x - 2 + 3x
This simplifies to:
5 >= x - 2
By adding 3x to both sides, we've successfully moved the '-3x' term from the left side to the right side, combining it with the '-2x' term. This step brings us closer to isolating 'x', making the inequality easier to solve. Grouping the 'x' terms is a fundamental technique in solving both equations and inequalities. It's about organizing the expression to make the variable the main focus.
3. Isolate 'x'
Our next goal is to isolate 'x' completely. We currently have 5 >= x - 2. To get 'x' by itself, we need to get rid of the '-2' on the right side. We can do this by adding 2 to both sides:
5 + 2 >= x - 2 + 2
This simplifies to:
7 >= x
Adding 2 to both sides cancels out the -2, leaving 'x' isolated. We now have 7 >= x, which means x is less than or equal to 7. This is a clear solution, telling us the range of values that x can take. Isolating the variable is the ultimate step in solving any inequality or equation. Once the variable is alone, we know its value or, in this case, the range of its possible values.
4. Rewrite the Solution
While 7 >= x is a perfectly valid solution, it's often clearer and more conventional to write it with 'x' on the left side. To do this, we simply flip the inequality, remembering to flip the direction of the inequality sign as well:
x <= 7
This reads as "x is less than or equal to 7." It's the same solution as 7 >= x, just expressed in a more common format. Rewriting the solution in this way makes it easier to understand the possible values of 'x' at a glance. It’s a matter of convention, but it helps in clear communication and avoids any potential confusion. So, our final solution is x <= 7, which means any value of x that is 7 or less will satisfy the original inequality.
Additional Discussion: Solving (3/4 + x) / (1/2)
The original request also included a second expression: (3/4 + x) / (1/2). This looks like another inequality or an expression that needs simplification. To clarify, let’s treat it as an expression that we need to simplify. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/2 is 2/1, which is simply 2. So, we multiply the expression (3/4 + x) by 2:
2 * (3/4 + x)
Now, we distribute the 2 across the terms inside the parentheses:
2 * (3/4) + 2 * x
This simplifies to:
3/2 + 2x
So, the simplified form of (3/4 + x) / (1/2) is 3/2 + 2x. If this was part of a larger inequality, we would use this simplified form in our calculations.
Combining Inequalities
If we were to combine this simplified expression with our previous inequality, we would need more context. For example, if the problem was:
(5 - 3x)/2 >= -x - 1 AND (3/4 + x) / (1/2) < some value
We would solve each inequality separately and then find the values of x that satisfy both. This often involves finding the intersection of the solution sets on a number line. However, without a specific inequality provided for the second expression, we can only simplify it.
Verification and Testing
To ensure our solution is correct, it's always a good idea to verify it. We can do this by plugging a value within our solution range (x <= 7) back into the original inequality. Let's pick x = 0, which is clearly less than or equal to 7:
(5 - 3(0))/2 >= -0 - 1
This simplifies to:
5/2 >= -1
Which is true, as 5/2 (or 2.5) is indeed greater than -1. This gives us confidence that our solution is correct. Now, let’s test a value outside our solution range, say x = 8:
(5 - 3(8))/2 >= -8 - 1
This simplifies to:
(5 - 24)/2 >= -9
-19/2 >= -9
-9.5 >= -9
This is false, as -9.5 is less than -9. This confirms that our solution x <= 7 is correct, as values outside this range do not satisfy the original inequality. Verification is a crucial step in solving inequalities (and equations). It helps catch any errors made during the solving process and ensures that the final answer is accurate.
Common Mistakes to Avoid
When solving inequalities, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them.
- Forgetting to Flip the Sign: As we mentioned earlier, the most common mistake is forgetting to flip the inequality sign when multiplying or dividing both sides by a negative number. Always double-check this step!
- Incorrectly Distributing: Make sure to distribute correctly when clearing parentheses. Each term inside the parentheses must be multiplied by the factor outside.
- Arithmetic Errors: Simple arithmetic mistakes can lead to incorrect solutions. Take your time and double-check your calculations, especially when dealing with fractions or negative numbers.
- Misinterpreting the Solution: Make sure you understand what your solution means. For example, x <= 7 means x can be any value less than or equal to 7, not just 7 itself.
- Not Verifying: Failing to verify your solution can lead to accepting an incorrect answer. Always plug a value from your solution range back into the original inequality to check.
Avoiding these common mistakes will greatly improve your accuracy and confidence in solving inequalities.
Real-World Applications
Inequalities aren't just abstract mathematical concepts; they have many practical applications in the real world. Let's explore a few examples:
- Budgeting: Inequalities are used to manage budgets. For instance, if you have a budget of $100 for groceries, and each item costs $5, you can use the inequality 5x <= 100 to determine the maximum number of items you can buy.
- Engineering: Engineers use inequalities to define safety margins and tolerances. For example, the load capacity of a bridge must be greater than the expected maximum load, which can be expressed as an inequality.
- Health and Fitness: Inequalities can be used to set fitness goals. For example, if you want to burn at least 500 calories during a workout, and each exercise burns 50 calories per minute, you can use the inequality 50x >= 500 to find out how many minutes you need to exercise.
- Business and Economics: Businesses use inequalities to model supply and demand, profit margins, and other economic factors. For example, a company might use an inequality to determine the minimum number of products they need to sell to break even.
- Computer Science: Inequalities are used in algorithms and data structures. For example, in sorting algorithms, inequalities are used to compare elements and determine their order.
These are just a few examples, but they illustrate how inequalities are a fundamental tool in various fields. Understanding inequalities can help you make informed decisions and solve problems in many aspects of life.
Conclusion
So, guys, we've successfully solved the inequality (5 - 3x)/2 >= -x - 1 and simplified the expression (3/4 + x) / (1/2). We've covered the step-by-step solution, discussed common mistakes to avoid, and explored real-world applications. Remember, the key to mastering inequalities is practice. Work through different problems, and don't hesitate to review the steps and concepts we've discussed. Inequalities are a crucial part of mathematics, and a solid understanding will benefit you in many areas. Keep practicing, and you'll become an inequality-solving pro in no time! Keep up the great work, and see you in the next math adventure!