Josefina's Stoppage Point Solving 3.5 = 1.9 - 0.8|2x - 0.6| Absolute Value Equation

Josefina's attempt to solve the equation $3.5 = 1.9 - 0.8|2x - 0.6|$ led her to a critical juncture at Step 3. Understanding why she stopped there requires a careful examination of the properties of absolute values and how they interact with algebraic equations. This article will delve into Josefina's steps, pinpoint the exact reason for her halt, and provide a comprehensive explanation for students tackling similar problems. We'll break down the concept of absolute value, explore its implications for equation-solving, and offer strategies for identifying when a solution is not possible.

Understanding the Steps

Let's meticulously analyze Josefina's progression through each step to understand her approach and identify the potential issue:

Step 1: Initial Equation

$$3. 5 = 1.9 - 0.8|2x - 0.6|$$

This is the starting point of the problem. Josefina is presented with an equation that involves an absolute value expression. The goal is to isolate x, but the absolute value introduces a complication. To effectively tackle this equation, it’s crucial to remember that the absolute value of a number represents its distance from zero, which is always non-negative. This inherent property of absolute values plays a crucial role in determining the possible solutions.

Step 2: Isolating the Absolute Value

$$1. 6 = -0.8|2x - 0.6|$$

In this step, Josefina subtracted 1.9 from both sides of the equation. This is a standard algebraic manipulation aimed at isolating the absolute value term. By subtracting 1.9, she simplified the equation and brought the absolute value expression closer to being by itself. This is a necessary step in solving for x because it allows for a clearer focus on the absolute value component and its influence on the solution.

To achieve this, she performed the subtraction: 3.5 - 1.9 = 1.6. This left her with the equation 1.6 = -0.8|2x - 0.6|. The next logical step would be to further isolate the absolute value by dividing both sides by -0.8.

Step 3: Dividing to Isolate Further

$$-2 = |2x - 0.6|$$

Here, Josefina divided both sides of the equation by -0.8. This step is crucial because it isolates the absolute value expression completely. By dividing, she aimed to get the absolute value term by itself on one side of the equation, which is essential for analyzing the possible solutions. The arithmetic is straightforward: 1.6 divided by -0.8 equals -2. This results in the equation -2 = |2x - 0.6|. This is where the problem arises, and we need to understand why.

The Critical Issue: Absolute Value Cannot Be Negative

The critical reason Josefina stopped at Step 3 lies in the fundamental definition of absolute value. The absolute value of any expression is, by definition, a non-negative quantity. It represents the distance of the expression from zero on the number line, and distance cannot be negative. This is a core concept in mathematics, and it's the key to understanding why Josefina's progression halted.

In Step 3, the equation is $-2 = |2x - 0.6|$. This equation states that the absolute value of the expression $(2x - 0.6)$ is equal to -2. This statement is inherently contradictory. Since the absolute value of any expression can never be negative, there is no value of x that can satisfy this equation. This is a crucial point to grasp when solving equations involving absolute values.

Why Josefina Stopped

Josefina stopped at Step 3 because she recognized the contradiction. The equation $-2 = |2x - 0.6|$ is mathematically impossible. There is no real number x that, when substituted into the expression, will result in an absolute value equal to a negative number. This understanding demonstrates a solid grasp of the properties of absolute values.

When students encounter such a situation, it's essential to recognize that the equation has no solution. Continuing to solve would be futile, as any further algebraic manipulations would still lead to a contradiction. The key takeaway here is that understanding the underlying mathematical principles, such as the non-negativity of absolute values, is crucial for effective problem-solving.

Implications and How to Recognize Such Situations

This situation highlights an important aspect of solving equations with absolute values: always check for contradictions. Here are some key implications and ways to recognize similar situations:

1. Isolate the Absolute Value: Before attempting to solve for the variable within the absolute value, always isolate the absolute value expression on one side of the equation. This helps to clearly see the value it is supposed to equal.
2. Check for Non-Negative Result: Once the absolute value is isolated, examine the other side of the equation. If it's negative, you can immediately conclude that there is no solution.
3. Understand the Definition: Reinforce the understanding that the absolute value represents distance from zero, which cannot be negative. This fundamental concept is crucial for solving absolute value equations.
4. Graphical Interpretation: Visualizing absolute value can also help. The graph of an absolute value function (e.g., y = |x|) is always above or on the x-axis, representing non-negative values. If you're trying to solve for where the absolute value equals a negative number, graphically, there's no intersection.

Alternative Scenarios and Solutions

To further illustrate the concept, let's consider a similar equation where a solution is possible:

Example:

$$3. 5 = 1.9 + 0.8|2x - 0.6|$$

Following the same steps as Josefina:

1. Subtract 1.9 from both sides:

$$1. 6 = 0.8|2x - 0.6|$$

2. Divide both sides by 0.8:

$$2 = |2x - 0.6|$$

In this scenario, the absolute value is equal to a positive number (2), which is perfectly valid. This means there are two possibilities to consider:

• Case 1: $2x - 0.6 = 2$
• Case 2: $2x - 0.6 = -2$

Each case leads to a different solution for x. Solving these linear equations:

• Case 1:

  $$2x = 2.6$$

  $$x = 1.3$$

• Case 2:

  $$2x = -1.4$$

  $$x = -0.7$$

Thus, in this example, there are two solutions: x = 1.3 and x = -0.7. This contrast highlights the importance of the sign on the other side of the equation after isolating the absolute value.

Conclusion

Josefina's decision to stop at Step 3 was the correct one. The equation $-2 = |2x - 0.6|$ has no solution because the absolute value of any expression cannot be negative. This example underscores the importance of understanding the fundamental properties of mathematical concepts, such as absolute value, when solving equations. By recognizing contradictions early on, students can save time and avoid fruitless efforts. Always remember to isolate the absolute value, check the sign on the other side of the equation, and recall the definition of absolute value as a non-negative quantity. This approach will help in effectively solving absolute value equations and identifying scenarios where no solution exists.