Evaluate Expressions With X = -1: Step-by-Step Solutions

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Evaluate Expressions with x = -1: Step-by-Step Solutions

Hey guys! Let's dive into a bit of algebra today. We're going to tackle evaluating some expressions when x is equal to -1. Don't worry, we'll break it down step by step so it's super easy to follow. This is a crucial skill in math, and mastering it now will help you big time later on. So, grab your pencils and let’s get started!

(i) Evaluating 2x - 7

In this first expression, 2x - 7, our goal is to find its value when x is -1. The key here is substitution. We're going to replace the variable x with the numerical value -1. It’s like swapping out a player in a game – x is leaving, and -1 is stepping onto the field! This is a fundamental concept in algebra and is used extensively when dealing with functions and equations.

  • Step 1: Substitute the value of x:

    This means we replace x with -1 in the expression. So, 2x becomes 2 * (-1). Remember, when a number is right next to a variable, it means multiplication. Pay close attention to the signs (positive and negative) because they are crucial in getting the correct answer. So, our expression now looks like this: 2*(-1) - 7.

  • Step 2: Perform the multiplication:

    Next, we multiply 2 by -1. A positive number multiplied by a negative number gives us a negative result. So, 2 * (-1) equals -2. Now our expression is simplified to -2 - 7. It’s all about breaking it down step by step, making it less intimidating and more manageable.

  • Step 3: Perform the subtraction:

    Now, we subtract 7 from -2. Think of it like owing someone $2 and then owing them another $7. In total, you owe $9, which we represent as -9. So, -2 - 7 = -9. And there you have it! The value of the expression 2x - 7 when x = -1 is -9. Remember to take your time, especially with negative numbers, and you'll nail it!

Therefore, the value of 2x - 7 when x = -1 is -9.

(ii) Evaluating -x + 2

Alright, let's move on to the second expression: -x + 2. This one is interesting because of the negative sign in front of the x. Remember, that negative sign is like an invisible -1 multiplying the x. So, -x is the same as -1 * x. Understanding this is crucial for correctly evaluating the expression. These seemingly small details can make a big difference in your final answer, so let’s pay close attention!

  • Step 1: Substitute the value of x:

    Just like before, we substitute x with -1. So, -x becomes -(-1). Notice how we have a negative sign outside the parentheses and a negative sign inside. This is where things can get a bit tricky, but don’t worry, we'll handle it. Now our expression looks like this: -(-1) + 2.

  • Step 2: Simplify the negative signs:

    A negative times a negative is a positive! So, -(-1) becomes +1. Think of it as the two negatives canceling each other out. This is a fundamental rule in math, and it’s super important to remember. Now our expression is even simpler: 1 + 2. See how we're making progress step by step?

  • Step 3: Perform the addition:

    Finally, we add 1 and 2. This is straightforward: 1 + 2 = 3. And that’s it! The value of the expression -x + 2 when x = -1 is 3. You’re doing awesome!

Thus, the value of -x + 2 when x = -1 is 3.

(iii) Evaluating x² + 2x + 1

Now, let's tackle a slightly more complex expression: x² + 2x + 1. This one involves an exponent (x squared) and multiple terms. But don't worry, we'll break it down just like before. The key here is to remember the order of operations (PEMDAS/BODMAS), which tells us to handle exponents before multiplication and addition. This expression is also a perfect example of a quadratic expression, which you'll encounter frequently in algebra.

  • Step 1: Substitute the value of x:

    We replace every instance of x with -1. So, x² becomes (-1)², and 2x becomes 2*(-1). Remember to put the -1 in parentheses, especially when dealing with exponents, to avoid sign errors. Our expression now looks like this: (-1)² + 2*(-1) + 1. See, not so scary when we take it one step at a time!

  • Step 2: Evaluate the exponent:

    (-1)² means -1 multiplied by itself: (-1) * (-1). A negative times a negative is a positive, so (-1)² = 1. Now our expression is 1 + 2*(-1) + 1. We're making good progress!

  • Step 3: Perform the multiplication:

    Next, we multiply 2 by -1, which gives us -2. So, our expression becomes 1 + (-2) + 1. Remember, adding a negative number is the same as subtracting the positive version of that number.

  • Step 4: Perform the addition and subtraction:

    Now we have 1 - 2 + 1. Let's do it step by step: 1 - 2 = -1, and then -1 + 1 = 0. So, the value of the expression is 0. You’ve got this!

Therefore, the value of x² + 2x + 1 when x = -1 is 0.

(iv) Evaluating 2x² - x - 2

Okay, let's move on to our final expression: 2x² - x - 2. This one combines exponents, multiplication, and subtraction, so it's a great way to practice everything we've learned so far. Just like with the previous expression, we need to remember the order of operations (PEMDAS/BODMAS) to make sure we solve it correctly. This is a crucial skill for tackling more complex algebraic problems.

  • Step 1: Substitute the value of x:

    We replace x with -1 in every instance. So, 2x² becomes 2*(-1)², and -x becomes -(-1). Our expression now looks like this: 2*(-1)² - (-1) - 2. Notice how we're carefully substituting and using parentheses where necessary. This helps prevent mistakes!

  • Step 2: Evaluate the exponent:

    As we saw before, (-1)² is (-1) * (-1), which equals 1. So, our expression becomes 2 * 1 - (-1) - 2. We're simplifying it step by step, making it much easier to handle.

  • Step 3: Perform the multiplication:

    Next, we multiply 2 by 1, which gives us 2. So, our expression is now 2 - (-1) - 2. Almost there!

  • Step 4: Simplify the negative signs:

    Remember, subtracting a negative is the same as adding a positive. So, -(-1) becomes +1. Our expression now reads 2 + 1 - 2.

  • Step 5: Perform the addition and subtraction:

    Let's do it step by step: 2 + 1 = 3, and then 3 - 2 = 1. So, the final value of the expression is 1. Awesome job!

Thus, the value of 2x² - x - 2 when x = -1 is 1.

Key Takeaways and Tips for Success

Great job working through these expressions with me! Remember, the key to success in algebra is to take things one step at a time. Here are some key takeaways and tips to help you master evaluating expressions:

  • Substitution is Key: Always start by substituting the given value for the variable. This is the foundation of evaluating expressions.
  • Order of Operations (PEMDAS/BODMAS): Remember Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is crucial for getting the correct answer.
  • Pay Attention to Signs: Negative signs can be tricky, so double-check your work and remember the rules for multiplying and dividing with negatives.
  • Use Parentheses: When substituting negative values, use parentheses to avoid sign errors. It's a simple trick that can save you a lot of headaches.
  • Break It Down: Complex expressions can seem intimidating, but if you break them down into smaller steps, they become much more manageable.
  • Practice Makes Perfect: The more you practice, the more comfortable you'll become with evaluating expressions. Try different examples and challenge yourself.

Conclusion

So there you have it, guys! We’ve successfully evaluated four different expressions for x = -1. Remember, algebra is like building with blocks – each concept builds on the previous one. Mastering these basic skills is crucial for tackling more advanced topics later on. Keep practicing, stay curious, and you’ll be an algebra whiz in no time! Keep up the great work, and I'll see you in the next math adventure!