Solving 2x - 3 > 11 - 5x: Find The Value Of X

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Solving 2x - 3 > 11 - 5x: Find the Value of x

Hey guys! Today, we are diving into a fun little math problem. We're going to figure out what values of x make the inequality 2x - 3 > 11 - 5x true. Inequalities might seem a bit tricky at first, but don't worry, we'll break it down step by step. Think of it like solving a puzzle where we need to find all the possible numbers that fit a certain rule. So, grab your thinking caps, and let's get started!

Understanding the Inequality

Before we jump into solving this, let's make sure we understand what the inequality 2x - 3 > 11 - 5x actually means. In simple terms, we're looking for any number x that, when plugged into the left side of the equation (2x - 3), gives us a result that is greater than what we get when we plug the same number into the right side (11 - 5x). It’s like a math seesaw, and we want the left side to be heavier than the right.

  • The variable in this case is x, which represents the unknown value we're trying to find. The goal is to isolate x on one side of the inequality, so we can see what values make the statement true.
  • The inequality symbol β€œ>” means β€œgreater than.” Remember, if we saw β€œ<,” that would mean β€œless than,” β€œβ‰₯” means β€œgreater than or equal to,” and β€œβ‰€β€ means β€œless than or equal to.”
  • The solution set will be a range of values, not just a single number. This is because inequalities often have many solutions. For example, if we find that x > 5, that means any number bigger than 5 will work.

Why Solve Inequalities?

You might be wondering, why do we even need to solve inequalities? Well, inequalities pop up in all sorts of real-world situations. Think about setting a budget (you can't spend more than your income), figuring out how many items you can buy with a certain amount of money, or even determining the safe range for a temperature in a science experiment. Understanding inequalities helps us make decisions and solve problems in everyday life.

Step-by-Step Solution

Okay, let's get down to business and solve this inequality! We'll follow a few key steps, just like we do when solving regular equations, but with a little twist or two.

1. Combine Like Terms

Our first goal is to get all the x terms on one side of the inequality and all the constant terms (the numbers without x) on the other side. To do this, we'll use addition and subtraction. The golden rule here is that whatever you do to one side of the inequality, you must do to the other side to keep things balanced.

So, let's start by getting rid of the -5x on the right side. We can do this by adding 5x to both sides:

2x - 3 + 5x > 11 - 5x + 5x

This simplifies to:

7x - 3 > 11

Now, let's move the -3 from the left side to the right side. We do this by adding 3 to both sides:

7x - 3 + 3 > 11 + 3

Which simplifies to:

7x > 14

2. Isolate the Variable

Great! We're almost there. Now we just need to get x by itself. Since x is being multiplied by 7, we need to do the opposite operation, which is division. We'll divide both sides of the inequality by 7:

(7x) / 7 > 14 / 7

This gives us:

x > 2

3. The Big Twist: Multiplying or Dividing by a Negative

Here's a crucial thing to remember when working with inequalities: If you multiply or divide both sides by a negative number, you have to flip the direction of the inequality sign. Why? Because multiplying or dividing by a negative number changes the order of the numbers on the number line. For example, 2 is greater than -3, but if we multiply both by -1, we get -2 and 3, and now 3 is greater than -2. So, the relationship flipped!

Luckily, in our problem, we divided by a positive number (7), so we don't need to worry about flipping the sign this time. But keep this in mind for future problems!

4. State the Solution

Okay, we've done it! We've found that x > 2. This is our solution. It means that any number greater than 2 will make the original inequality true. Easy peasy, right?

Understanding the Solution Set

So, we know that x > 2, but what does that really mean? Let's think about it in terms of a number line. Imagine a line stretching out infinitely in both directions, with zero in the middle. The numbers get bigger as we move to the right and smaller as we move to the left.

Our solution, x > 2, means we're interested in all the numbers to the right of 2 on the number line. But what about 2 itself? Does 2 count as a solution? Well, our inequality says x is greater than 2, not greater than or equal to 2. So, 2 itself is not included.

Representing the Solution Set

There are a couple of ways we can represent this solution set:

  • Graphically: On a number line, we would draw an open circle at 2 (to show that 2 is not included) and then draw an arrow extending to the right, shading in all the numbers greater than 2.
  • Interval Notation: This is a handy way to write the solution set using parentheses and brackets. For x > 2, we would write the solution as (2, ∞). The parenthesis next to the 2 means 2 is not included, and the ∞ (infinity symbol) means the solution goes on forever to the right. If we had x β‰₯ 2, we would use a bracket instead: [2, ∞), which means 2 is included.

Testing Our Solution

It's always a good idea to double-check our work, especially with inequalities. We can do this by picking a number from our solution set (a number greater than 2) and plugging it back into the original inequality to see if it makes the statement true.

Let's pick a number like 3. If we plug 3 into the original inequality, 2x - 3 > 11 - 5x, we get:

2(3) - 3 > 11 - 5(3)

6 - 3 > 11 - 15

3 > -4

This is true! 3 is greater than -4, so our solution set seems to be correct. We can also pick a number not in our solution set (like 1) and plug it in. You'll see that it makes the inequality false, which further confirms our answer.

Common Mistakes to Avoid

Inequalities are pretty straightforward, but there are a few common pitfalls to watch out for:

  • Forgetting to Flip the Sign: This is the big one! Remember, if you multiply or divide by a negative number, you must flip the inequality sign. If you don't, you'll get the wrong solution.
  • Treating Inequalities Like Equations: While the steps for solving inequalities are similar to those for equations, it's important to remember that inequalities represent a range of solutions, not just a single value. Don't stop at just finding one number that works; think about the whole set of possibilities.
  • Misinterpreting the Inequality Symbol: Make sure you know what each symbol means. β€œ>” is greater than, β€œ<” is less than, β€œβ‰₯” is greater than or equal to, and β€œβ‰€β€ is less than or equal to. Getting these mixed up can lead to incorrect solutions.

Real-World Applications

We talked a bit about why inequalities are important, but let's look at a few more specific examples of how they're used in the real world:

  • Budgeting: Imagine you have $50 to spend at the store. You can use an inequality to figure out how many items you can buy if each item costs a certain amount. For example, if each item costs $5, you could set up the inequality 5x ≀ 50 (where x is the number of items) and solve for x to find the maximum number of items you can purchase.
  • Speed Limits: Speed limits on roads are a great example of inequalities in action. The sign might say the speed limit is 65 mph, which means you can drive up to 65 mph, but not faster. This can be expressed as x ≀ 65, where x is your speed.
  • Healthy Ranges: Doctors often use inequalities to define healthy ranges for things like blood pressure, cholesterol levels, and blood sugar. For example, a healthy blood pressure might be defined as less than 120/80, which can be written as two inequalities: systolic pressure < 120 and diastolic pressure < 80.
  • Manufacturing: In manufacturing, inequalities are used to set tolerances for product dimensions. For example, a part might need to be within a certain range of sizes to function correctly. If the ideal size is 10 cm, the tolerance might be Β±0.1 cm, which means the part needs to be between 9.9 cm and 10.1 cm. This can be expressed as 9.9 ≀ x ≀ 10.1, where x is the actual size of the part.

Practice Problems

Now that we've gone through the steps and looked at some examples, it's time to practice! Here are a few more inequalities for you to try solving on your own:

  1. 3x + 5 < 14
  2. -2x - 1 β‰₯ 7
  3. 4x - 6 > 2x + 8

Remember to follow the steps we discussed: combine like terms, isolate the variable, and don't forget to flip the sign if you multiply or divide by a negative number. Grab a pencil and paper, and give these a shot! Solving inequalities is like riding a bike – it might seem wobbly at first, but the more you practice, the easier it gets.

Conclusion

Alright, awesome job guys! We've successfully tackled the inequality 2x - 3 > 11 - 5x and found that the solution is x > 2. We've also learned how to represent the solution set, avoid common mistakes, and see how inequalities pop up in the real world. Inequalities are a super useful tool in math and in life, so mastering them is definitely worth the effort.

Keep practicing, and you'll be solving inequalities like a pro in no time! And remember, math can be fun – especially when you break it down step by step. Until next time, happy problem-solving!