Solving Absolute Value Equations 5 = |3b - 4| For B

by Kenji Nakamura 52 views

Hey guys! Today, we're diving into a fun little math problem where we need to solve for the variable b in an equation involving absolute values. Don't worry, it's not as scary as it sounds! We'll break it down step by step, making sure everyone understands the process. Our equation is 5 = |3b - 4|, and we've got some multiple-choice options for the possible values of b. Let's get started!

Understanding Absolute Value

Before we jump into solving, let's quickly recap what absolute value means. The absolute value of a number is its distance from zero on the number line. Think of it as the magnitude of the number, regardless of whether it's positive or negative. For instance, the absolute value of 3 (written as |3|) is 3, and the absolute value of -3 (written as |-3|) is also 3. This is because both 3 and -3 are three units away from zero.

Now, when we have an equation like 5 = |3b - 4|, it means that the expression inside the absolute value, which is (3b - 4), can be either 5 or -5. This is the key to solving these types of problems. We need to consider both possibilities to find all possible values of b.

Breaking Down the Equation: Two Possibilities

So, because of the absolute value, we have two scenarios to explore:

  1. The expression inside the absolute value, (3b - 4), equals 5.
  2. The expression inside the absolute value, (3b - 4), equals -5.

Let's tackle each of these scenarios one at a time.

Scenario 1: 3b - 4 = 5

In this case, we're assuming that the expression (3b - 4) is equal to 5. This gives us a simple linear equation to solve for b. Here's how we do it:

  • Step 1: Isolate the term with b. To do this, we need to get rid of the -4 on the left side of the equation. We can do this by adding 4 to both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced.

    3b - 4 + 4 = 5 + 4

    This simplifies to:

    3b = 9

  • Step 2: Solve for b. Now, we have 3b equals 9. To find b, we need to get it by itself. We can do this by dividing both sides of the equation by 3:

    3b / 3 = 9 / 3

    This simplifies to:

    b = 3

    So, one possible value for b is 3. Keep this in mind as we move on to the second scenario.

Scenario 2: 3b - 4 = -5

Now, let's consider the second possibility, where the expression (3b - 4) is equal to -5. Again, we have another linear equation to solve:

  • Step 1: Isolate the term with b. Just like before, we need to get rid of the -4 on the left side. We add 4 to both sides:

    3b - 4 + 4 = -5 + 4

    This simplifies to:

    3b = -1

  • Step 2: Solve for b. To isolate b, we divide both sides by 3:

    3b / 3 = -1 / 3

    This simplifies to:

    b = -1/3

    So, our second possible value for b is -1/3.

Putting It All Together: The Solutions for b

We've explored both scenarios and found two possible values for b: 3 and -1/3. This means that if we plug either of these values back into the original equation, 5 = |3b - 4|, the equation will hold true. Let's quickly check our work to be sure.

  • Checking b = 3:

    5 = |3(3) - 4|

    5 = |9 - 4|

    5 = |5|

    5 = 5 (This is true!)

  • Checking b = -1/3:

    5 = |3(-1/3) - 4|

    5 = |-1 - 4|

    5 = |-5|

    5 = 5 (This is also true!)

Both values work, so we've successfully solved for b.

Choosing the Correct Answer

Now that we know the possible values of b are 3 and -1/3, we can look back at our multiple-choice options and find the one that matches. The correct answer is B. b = 3 and b = -1/3.

Key Takeaways for Solving Absolute Value Equations

Let's recap the key steps for solving equations involving absolute values:

  1. Understand Absolute Value: Remember that the absolute value of a number is its distance from zero, so it can be either positive or negative.
  2. Create Two Scenarios: When you have an equation with an absolute value, set up two equations: one where the expression inside the absolute value equals the positive value on the other side of the equation, and one where it equals the negative value.
  3. Solve Each Equation: Solve each of the resulting equations separately using standard algebraic techniques.
  4. Check Your Answers: Always plug your solutions back into the original equation to make sure they work.
  5. Choose the Correct Option: Select the solution that satisfies the original equation from the multiple choices.

Common Mistakes to Avoid

  • Forgetting the Negative Case: The most common mistake is only considering the positive case of the absolute value and forgetting to set up a second equation for the negative case. This will lead to missing one of the solutions.
  • Incorrectly Distributing: Be careful when distributing if there's a coefficient outside the absolute value. Make sure you're applying the operations in the correct order.
  • Not Checking Solutions: Always, always, always check your solutions! This is the best way to catch any errors you might have made along the way.

Practice Problems

Want to test your understanding? Try solving these similar absolute value equations:

  1. |2x + 1| = 7
  2. |4y - 3| = 5
  3. |z + 6| = 2

Solving these problems will help solidify your understanding of the process and build your confidence in tackling these types of questions. You can do it, guys!

Conclusion

So, there you have it! We've successfully solved for b in the equation 5 = |3b - 4| and learned the ins and outs of dealing with absolute value equations. Remember to break down the problem into two scenarios, solve each one carefully, and always check your answers. With a little practice, you'll become absolute value equation-solving pros! Keep up the great work, and happy problem-solving!