Absolute Value Equation: Why Josefina Stopped Solving

Hey guys! Let's dive into a common head-scratcher in algebra: absolute value equations. We're going to break down a specific example and see why a student, Josefina, hit a roadblock. Understanding this will help you tackle similar problems with confidence. So, grab your pencils, and let's get started!

Josefina's Attempt

Josefina was on a mission to solve for x in the absolute value equation:

${3.5 = 1.9 - 0.8|2x - 0.6|}$

Here's how she approached it:

1. ${3.5 = 1.9 - 0.8|2x - 0.6|}$
2. ${1.6 = -0.8|2x - 0.6|}$
3. ${-2 = |2x - 0.6|}$

But then, she stopped. The big question is: Why did Josefina stop at this point? Let's dissect this step-by-step to figure out what went wrong and why there was the need to stop the process.

Understanding Absolute Value

Before we jump to conclusions, let's quickly refresh our understanding of what absolute value really means. The absolute value of a number is its distance from zero, always a non-negative value. Think of it like this: whether you're at +5 or -5 on the number line, you're still 5 units away from zero. So, ${|5| = 5}$ and ${|-5| = 5}$. This is a critical concept to keep in mind when solving equations involving absolute values.

Spotting the Problem

Okay, with the definition of absolute value fresh in our minds, let's revisit Josefina's last step:

${-2 = |2x - 0.6|}$

Here's the key observation: the absolute value of anything can never be negative. The left side of the equation is -2, which is negative, while the right side is an absolute value, which must be non-negative. This is a fundamental contradiction!

In simpler terms:

• Absolute value always results in a positive number or zero.
• We have an absolute value equal to a negative number.
• This is impossible!

Because of this contradiction, Josefina correctly recognized that there was no point in continuing. The equation has no solution. Any further steps would be futile because the equation itself is flawed at this stage.

Why Josefina Stopped: The Explanation

The reason Josefina stopped is that she reached a point where the equation presented an impossible scenario: an absolute value equal to a negative number. Since the absolute value of any expression is always non-negative (i.e., zero or positive), the equation ${-2 = |2x - 0.6|}$ has no solution. There's no value of x that can make this equation true. Therefore, there was no point in continuing the solving process. Recognizing such contradictions is a crucial skill in algebra. It prevents you from wasting time on equations that have no solutions and helps you develop a deeper understanding of the properties of mathematical operations.

Key Takeaways

• Absolute Value is Always Non-Negative: The absolute value of any expression will always be greater than or equal to zero.
• Look for Contradictions: When solving equations, be alert for situations that violate fundamental mathematical principles.
• No Solution: If you encounter a contradiction, it often means the equation has no solution.
• Save Time: Recognizing contradictions early can save you from unnecessary work.

Solving Absolute Value Equations: A General Approach

Now that we've dissected why Josefina stopped, let's briefly discuss the general approach to solving absolute value equations when they do have solutions. The key is to remember that the expression inside the absolute value can be either positive or negative.

Here's the general strategy:

1. Isolate the Absolute Value: Get the absolute value expression by itself on one side of the equation.
2. Split into Two Equations: Create two separate equations:
   • One where the expression inside the absolute value is equal to the positive value on the other side of the equation.
   • One where the expression inside the absolute value is equal to the negative value on the other side of the equation.
3. Solve Each Equation: Solve each of the two equations you created in step 2.
4. Check Your Solutions: It's crucial to plug your solutions back into the original equation to make sure they are valid. Sometimes, you might get extraneous solutions (solutions that don't actually work).

Example:

Let's say you have the equation ${|x - 3| = 5}$. Here's how you'd solve it:

1. Isolate: The absolute value is already isolated.
2. Split:
   • ${x - 3 = 5}$
   • ${x - 3 = -5}$
3. Solve:
   • ${x = 8}$
   • ${x = -2}$
4. Check:
   • ${|8 - 3| = |5| = 5}$ (Valid)
   • ${|-2 - 3| = |-5| = 5}$ (Valid)

So, the solutions are ${x = 8}$ and ${x = -2}$.

Let's Practice!

To solidify your understanding, try solving these absolute value equations:

1. ${|2x + 1| = 7}$
2. ${3|x - 4| = 9}$
3. ${|5 - x| = 2}$

Remember to follow the steps we discussed, and always check your solutions!

Conclusion

Josefina's experience highlights a crucial aspect of solving absolute value equations: recognizing when a solution is impossible. By understanding the fundamental property that absolute values are always non-negative, you can avoid wasting time on equations that have no solution. Keep practicing, and you'll become a pro at solving absolute value equations! Remember, math isn't just about finding answers; it's about understanding the underlying principles and recognizing patterns. Now go out there and conquer those equations!