Solving Inequalities A Step-by-Step Guide To Solve For Q In 50q + 43 > -11q + 70

by Marta Kowalska 81 views

Hey guys! Let's dive into solving inequalities, specifically when we're looking for the value of 'q'. Inequalities might seem a bit tricky at first, but they're super manageable once you break them down step by step. In this guide, we'll tackle an example inequality, walk through each step involved in finding the solution, and make sure we reduce fractions to their simplest forms along the way. No rounding or mixed fractions here – we're going to keep things precise and clean. So, grab your pencils, and let's get started!

Understanding Inequalities

Before we jump into solving, let's quickly recap what inequalities are all about. Unlike equations that use an equals sign (=), inequalities use signs like > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to). These signs help us express relationships where two values aren't necessarily equal but one is larger or smaller than the other. Inequalities often have a range of solutions rather than a single answer, which adds a fun twist to problem-solving. Think of it like finding all the numbers that fit a certain condition, rather than just one magic number.

In the inequality we're going to solve, 50q + 43 > -11q + 70, our goal is to isolate 'q' on one side of the inequality. This will tell us all the values that 'q' can be for the inequality to hold true. We'll use similar techniques as solving equations, like adding, subtracting, multiplying, and dividing, but we need to remember one crucial rule: when we multiply or divide both sides by a negative number, we need to flip the inequality sign. Keep this in mind, and we'll be golden!

The Given Inequality: 50q + 43 > -11q + 70

Alright, let's break down the inequality we're going to solve: 50q + 43 > -11q + 70. This might look a bit intimidating at first glance, but don't worry, we're going to tackle it piece by piece. Our main goal here is to isolate 'q' on one side of the inequality. This means we want to get all the terms with 'q' on one side and all the constant terms (the numbers) on the other side. Think of it like sorting your laundry – we're going to group similar items together.

To start, we need to decide which side we want to put our 'q' terms on. A good strategy is to move the term with the smaller coefficient (the number in front of 'q') to the side with the larger coefficient. In this case, we have 50q on the left and -11q on the right. Since -11 is smaller than 50, we'll want to move the -11q term to the left side. This will help us avoid dealing with negative coefficients later on, which can sometimes make things a bit trickier. So, let's get moving!

Step 1: Combining 'q' Terms

Okay, let's get those 'q' terms together! We have 50q + 43 > -11q + 70, and we want to move the -11q from the right side to the left side. To do this, we'll use the good old addition trick. Remember, whatever we do to one side of the inequality, we have to do to the other side to keep things balanced. So, we're going to add 11q to both sides of the inequality.

Adding 11q to both sides gives us:

(50q + 43) + 11q > (-11q + 70) + 11q

Now, let's simplify each side. On the left side, we have 50q + 11q, which combines to 61q. So the left side becomes 61q + 43. On the right side, we have -11q + 11q, which cancels each other out, leaving us with just 70. Our inequality now looks like this:

61q + 43 > 70

See? We're making progress! We've successfully moved all the 'q' terms to one side. Now, we need to deal with the constant terms and get them over to the other side. Let's move on to step two!

Step 2: Isolating the 'q' Term

Great job on combining the 'q' terms! Now, let's focus on isolating the 'q' term completely. We've got 61q + 43 > 70, and we need to get rid of that +43 on the left side. To do this, we'll use the opposite operation – subtraction. We're going to subtract 43 from both sides of the inequality.

Subtracting 43 from both sides gives us:

(61q + 43) - 43 > 70 - 43

Now, let's simplify. On the left side, +43 and -43 cancel each other out, leaving us with just 61q. On the right side, 70 - 43 equals 27. So our inequality now looks like this:

61q > 27

We're almost there! We've managed to isolate the 'q' term on one side. Now, we just need to get 'q' by itself by dealing with the coefficient 61. This is where division comes in handy. Let's move on to the final step!

Step 3: Solving for 'q'

Alright, we've reached the final step in solving for 'q'! We've got the inequality 61q > 27, and we need to get 'q' all by itself. Currently, 'q' is being multiplied by 61, so to undo this multiplication, we're going to divide both sides of the inequality by 61.

Dividing both sides by 61 gives us:

(61q) / 61 > 27 / 61

Now, let's simplify. On the left side, 61q divided by 61 is simply 'q'. On the right side, we have the fraction 27/61. Now, we need to check if this fraction can be reduced to its lowest terms. To do this, we look for any common factors between the numerator (27) and the denominator (61). The factors of 27 are 1, 3, 9, and 27. The number 61 is a prime number, which means its only factors are 1 and itself. Since 27 and 61 have no common factors other than 1, the fraction 27/61 is already in its lowest terms.

So, our final inequality is:

q > 27/61

And there you have it! We've successfully solved for 'q'. This means that 'q' can be any number greater than 27/61 for the original inequality to hold true. We didn't round our answer, and we made sure our fraction was in its simplest form. High five!

Final Answer

So, after all the steps, the solution to the inequality 50q + 43 > -11q + 70 is:

q > 27/61

This tells us that any value of 'q' greater than 27/61 will satisfy the original inequality. Remember, the key to solving inequalities is to treat them much like equations, but with the important rule of flipping the inequality sign if you multiply or divide by a negative number. We've walked through each step carefully, making sure to simplify along the way. You've got this!

Key Takeaways

Before we wrap up, let's quickly recap the key steps we took to solve this inequality. This will help solidify your understanding and give you a handy reference for tackling similar problems in the future. Remember, practice makes perfect, so the more you work with inequalities, the more comfortable you'll become.

  1. Combine 'q' Terms: The first step is to gather all the terms containing 'q' on one side of the inequality. We do this by adding or subtracting terms from both sides to move them where we want them. In our example, we added 11q to both sides to get the 'q' terms on the left.
  2. Isolate the 'q' Term: Next, we want to isolate the 'q' term by getting rid of any constant terms on the same side. We do this by performing the opposite operation. In our case, we subtracted 43 from both sides to isolate 61q.
  3. Solve for 'q': Finally, we solve for 'q' by dividing both sides of the inequality by the coefficient of 'q'. Remember to flip the inequality sign if you're dividing by a negative number (we didn't have to do that in this example). We then simplified the resulting fraction to its lowest terms.

And that's it! By following these steps, you can confidently solve a wide range of inequalities. Keep practicing, and you'll become an inequality-solving pro in no time!

Tips for Success

Solving inequalities might seem daunting at first, but with a few tips and tricks up your sleeve, you'll be acing them in no time. Here are some helpful hints to keep in mind as you work through inequality problems:

  • Always remember to flip the inequality sign when you multiply or divide both sides by a negative number. This is a crucial step that's easy to overlook, so make it a habit to double-check whenever you perform these operations.
  • It's often easier to avoid dealing with negative coefficients by strategically moving terms around. For example, if you have -3q on one side and 5q on the other, move the -3q to the side with the 5q. This will help you avoid flipping the inequality sign unnecessarily.
  • Simplify fractions to their lowest terms. This not only gives you the most accurate answer but also makes it easier to compare and work with your results.
  • Check your answer! A great way to make sure you've solved the inequality correctly is to pick a value that fits your solution and plug it back into the original inequality. If the inequality holds true, you're on the right track.

By keeping these tips in mind and practicing regularly, you'll build your confidence and skills in solving inequalities. Keep up the great work!

Practice Problems

Now that we've walked through an example and discussed some helpful tips, it's time to put your skills to the test! Here are a few practice problems that you can try solving on your own. Remember to follow the steps we've outlined and pay close attention to the details.

  1. Solve for x: 3x - 5 < 7x + 3
  2. Solve for y: -2(y + 4) ≥ 6y - 12
  3. Solve for z: 4z + 9 > 2z - 1

Take your time, work through each problem step by step, and don't be afraid to make mistakes – that's how we learn! Once you've solved the problems, you can check your answers with online resources or ask a friend or teacher for help. The more you practice, the better you'll become at solving inequalities. Happy solving!

Conclusion

Well, guys, we've reached the end of our journey into solving inequalities! We've covered a lot of ground, from understanding the basics of inequalities to tackling a step-by-step example and exploring some helpful tips and tricks. Remember, the key to success in math, as in life, is practice. The more you work with inequalities, the more confident and skilled you'll become.

We started by breaking down the inequality 50q + 43 > -11q + 70, and we systematically worked through each step to isolate 'q'. We combined 'q' terms, isolated the 'q' term itself, and finally solved for 'q', arriving at the solution q > 27/61. We also emphasized the importance of simplifying fractions and remembering to flip the inequality sign when multiplying or dividing by a negative number.

So, keep practicing, stay curious, and don't be afraid to ask for help when you need it. You've got this! And who knows, maybe inequalities will become your new favorite math topic. Until next time, happy solving!