Solving Inequalities: Find F / -1 + 5 ≤ 9

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Hey guys! Today, we're diving into the world of inequalities and tackling a specific problem: solving for f in the inequality f / -1 + 5 ≤ 9. Don't worry, it sounds more complicated than it actually is. We'll break it down step-by-step so you can confidently solve similar problems in the future. Let's get started!

Understanding Inequalities

Before we jump into the problem, let's quickly recap what inequalities are. Unlike equations, which have a single solution, inequalities represent a range of possible solutions. Think of it like this: an equation is a precise balance, while an inequality is more like a seesaw with a little wiggle room. Common inequality symbols include:

  • < (less than)
  • > (greater than)
  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)

Our goal is to isolate f on one side of the inequality, just like we would with an equation. The key difference is that we need to be mindful of what happens when we multiply or divide by a negative number, but we'll get to that shortly. Remember, inequalities are all about finding the range of values that make the statement true. This isn't just about one single answer; it's about a whole set of possibilities!

Step-by-Step Solution

Okay, let's get to the problem at hand: f / -1 + 5 ≤ 9. Here's how we'll solve it:

1. Isolate the Term with f

Our first step is to get the term with f by itself on one side of the inequality. To do this, we need to get rid of the + 5. We can do this by subtracting 5 from both sides of the inequality. Remember, whatever we do to one side, we must do to the other to keep the inequality balanced. This is a fundamental principle in solving both equations and inequalities.

f / -1 + 5 ≤ 9

Subtract 5 from both sides:

f / -1 + 5 - 5 ≤ 9 - 5

This simplifies to:

f / -1 ≤ 4

2. Multiply by -1 (and Remember the Flip!)

Now, we need to get f by itself. It's currently being divided by -1. To undo this division, we need to multiply both sides of the inequality by -1. This is where the crucial rule comes into play: When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign. This is because multiplying or dividing by a negative number changes the direction of the inequality.

(f / -1) * (-1) ≥ 4 * (-1)

Notice that we've flipped the ≤ sign to ≥. This is absolutely critical for getting the correct solution. Forgetting to flip the sign is a very common mistake, so always double-check when you're working with negative numbers.

3. Simplify

After multiplying, we simplify:

f ≥ -4

And that's it! We've solved for f. The solution is f is greater than or equal to -4.

Understanding the Solution

The solution f ≥ -4 means that any value of f that is greater than or equal to -4 will satisfy the original inequality. This isn't just one number; it's a whole range of numbers. For example, -4, -3, -2, 0, 1, 10, and 100 all work. Let's test a couple of them to see:

  • If f = -4: (-4) / -1 + 5 = 4 + 5 = 9. This satisfies the inequality because 9 ≤ 9.
  • If f = 0: (0) / -1 + 5 = 0 + 5 = 5. This satisfies the inequality because 5 ≤ 9.
  • If f = -5: (-5) / -1 + 5 = 5 + 5 = 10. This does not satisfy the inequality because 10 is not less than or equal to 9.

This reinforces the idea that f must be greater than or equal to -4.

Common Mistakes and How to Avoid Them

Solving inequalities is pretty straightforward once you understand the rules, but there are a couple of common mistakes that students often make. Let's talk about them so you can avoid them:

1. Forgetting to Flip the Inequality Sign

This is, without a doubt, the most common mistake. As we emphasized earlier, you must flip the inequality sign whenever you multiply or divide both sides by a negative number. Make it a habit to double-check this step whenever you're working with negative numbers. A good trick is to circle the inequality sign whenever you perform this operation to remind yourself that you've flipped it.

2. Incorrectly Applying the Order of Operations

Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? Make sure you're following the correct order of operations when simplifying the inequality. In our example, we first isolated the term with f by subtracting 5. Then, we dealt with the division by -1.

3. Not Understanding the Solution Set

It's important to understand that the solution to an inequality is a set of numbers, not just a single number. Take some time to think about what your solution means. Does it make sense in the context of the problem? Try plugging in a few values to check if they satisfy the inequality.

Practice Problems

Now that we've walked through the solution and discussed common mistakes, it's time to put your knowledge to the test! Here are a few practice problems for you to try:

  1. Solve for x: x / -2 + 3 > 7
  2. Solve for y: -3y + 1 ≤ 10
  3. Solve for z: 4 - z / 5 ≥ 2

Work through these problems carefully, paying close attention to the rule about flipping the inequality sign. Remember to show your work so you can track your steps and identify any errors.

Conclusion

So, there you have it! We've successfully solved the inequality f / -1 + 5 ≤ 9. We've learned how to isolate the variable, the importance of flipping the inequality sign when multiplying or dividing by a negative number, and how to interpret the solution set. Inequalities are a fundamental concept in mathematics, and mastering them will be incredibly beneficial as you progress in your studies. Keep practicing, and you'll become a pro in no time!

Remember, math isn't about memorizing formulas; it's about understanding the concepts and developing problem-solving skills. So, keep exploring, keep questioning, and keep learning! You got this!