Solving Absolute Value Equations - Determining The True Statement
#seo-title: Solving Absolute Value Equations - Which Statement is True?
Determining the truthfulness of statements involving absolute value equations requires a systematic approach. We need to understand the properties of absolute values and how they affect the solutions of equations. This article will delve into each given equation, dissecting the process of solving them and validating the proposed number of solutions. By carefully examining the nature of absolute value expressions and their behavior under different conditions, we can accurately determine which statement holds true. This exploration will not only reinforce your understanding of absolute value equations but also equip you with the skills to solve similar problems with confidence.
Understanding Absolute Value
Before we dive into the specific equations, let's clarify the fundamental concept of absolute value. The absolute value of a number is its distance from zero on the number line. It is always non-negative. Mathematically, we denote the absolute value of x as |x|. This means:
- If x ≥ 0, then |x| = x
- If x < 0, then |x| = -x
This definition is crucial because it tells us that an absolute value expression will always yield a non-negative result. This non-negativity is the key to understanding when an absolute value equation might have no solution, one solution, or two solutions. When solving absolute value equations, we often need to consider two cases: the expression inside the absolute value is positive or zero, and the expression inside the absolute value is negative.
Implications for Solving Equations
The understanding of absolute value's non-negativity is pivotal when assessing the possibility of solutions in absolute value equations. Consider an equation of the form |expression| = constant. If the constant is negative, there will be no solution because the absolute value of any expression cannot be negative. Conversely, if the constant is zero, there will be at most one solution, found by setting the expression inside the absolute value bars equal to zero. If the constant is positive, we typically have two potential solutions, arising from the two cases mentioned earlier. One case occurs when the expression inside the absolute value is equal to the positive constant, and another when it's equal to the negative constant. This bifurcation in solutions stems directly from the definition of absolute value, where a negative input is turned positive.
When approaching absolute value equations, the first step often involves isolating the absolute value expression on one side of the equation. This simplification makes it easier to assess the nature of potential solutions, as it directly reveals the value that the absolute value expression must equal. Once the absolute value expression is isolated, we can then apply the definition of absolute value to systematically explore the cases, leading to a comprehensive understanding of all possible solutions. Therefore, recognizing the inherent properties of absolute values is not merely a conceptual understanding but a fundamental skill required for efficiently and accurately solving equations involving absolute values.
Analyzing the Equations
Now, let's analyze each equation provided in the problem to determine which statement is true. We will systematically solve each equation and check the number of solutions.
Equation 1: -3|2x + 1.2| = -1
To analyze the equation -3|2x + 1.2| = -1, we must first isolate the absolute value term. This involves dividing both sides of the equation by -3, which yields |2x + 1.2| = 1/3. Now that we have isolated the absolute value, we can consider the two cases derived from the definition of absolute value. The first case arises when the expression inside the absolute value is positive or zero, and the second case occurs when the expression inside the absolute value is negative.
For the first case, 2x + 1.2 = 1/3, we solve for x by first subtracting 1.2 from both sides, converting 1.2 into a fraction to easily compute the difference: 2x = 1/3 - 1.2 = 1/3 - 6/5. To combine these fractions, we find a common denominator, which is 15, leading to 2x = (5 - 18)/15 = -13/15. Dividing both sides by 2 gives x = -13/30. This is a potential solution, and it satisfies the condition of the first case.
Next, we consider the second case where 2x + 1.2 is negative, which means -(2x + 1.2) = 1/3. Distributing the negative sign, we get -2x - 1.2 = 1/3. Adding 1.2 to both sides (again converting to a fraction) yields -2x = 1/3 + 1.2 = 1/3 + 6/5. Using the common denominator of 15, we have -2x = (5 + 18)/15 = 23/15. Dividing by -2, we find x = -23/30. This also gives a valid solution, as it satisfies the second case's condition.
Thus, the equation -3|2x + 1.2| = -1 has two solutions. Given that the statement claims there is no solution, it is incorrect. The detailed steps in solving this equation demonstrate the methodology for handling absolute value equations, emphasizing the importance of considering both positive and negative cases when the absolute value is isolated and set equal to a positive number.
Equation 2: 3.5|6x - 2| = 3.5
To determine the number of solutions for the equation 3.5|6x - 2| = 3.5, we start by isolating the absolute value term. Dividing both sides of the equation by 3.5 yields |6x - 2| = 1. Now, we apply the fundamental principle of absolute values, which dictates that we must consider two cases: one where the expression inside the absolute value is equal to the positive value (1), and another where the expression is equal to the negative value (-1).
In the first case, 6x - 2 = 1, we solve for x by adding 2 to both sides, which gives 6x = 3. Then, dividing both sides by 6, we find x = 3/6 = 1/2. This is a potential solution, and we verify its validity by substituting it back into the original equation. Doing so confirms that x = 1/2 is indeed a solution.
For the second case, 6x - 2 = -1, we again solve for x by adding 2 to both sides, which results in 6x = 1. Dividing both sides by 6, we obtain x = 1/6. Similar to the first case, we verify this solution by substituting x = 1/6 back into the original equation. This substitution confirms that x = 1/6 is also a solution.
Therefore, the equation 3.5|6x - 2| = 3.5 has two solutions: x = 1/2 and x = 1/6. This contradicts the statement that the equation has only one solution. The process of solving this equation underscores the importance of systematically addressing both positive and negative cases of the absolute value expression, ensuring that all possible solutions are identified and verified.
Equation 3: 5|-3.1x + 6.9| = -3.5
The equation 5|-3.1x + 6.9| = -3.5 presents a unique scenario that allows us to quickly determine the nature of its solutions. The first crucial step in analyzing this equation is to isolate the absolute value term. This is achieved by dividing both sides of the equation by 5, which yields |-3.1x + 6.9| = -3.5/5 = -0.7.
At this point, a fundamental property of absolute values comes into play. The absolute value of any expression, regardless of the expression's composition, is always non-negative. This is because the absolute value represents the distance from zero, which cannot be negative. Therefore, an absolute value expression can never be equal to a negative number. In our case, we have the absolute value of an expression, |-3.1x + 6.9|, equated to a negative value, -0.7.
This direct contradiction immediately indicates that the equation has no solution. There is no value of x that can satisfy this equation because the left-hand side, an absolute value, will always be non-negative, while the right-hand side is negative. This understanding is critical in solving absolute value equations as it provides a quick check for the feasibility of solutions. Recognizing this contradiction early in the process saves time and effort, preventing the need to explore cases that are mathematically impossible.
Thus, the statement that the equation 5|-3.1x + 6.9| = -3.5 has two solutions is definitively false. The equation has no solution due to the absolute value being equated to a negative number, a mathematical impossibility.
Equation 4: -0.3|3 + 8x| = 0.9
To analyze the equation -0.3|3 + 8x| = 0.9, the initial step, as with other absolute value equations, is to isolate the absolute value term. We achieve this by dividing both sides of the equation by -0.3. This operation results in |3 + 8x| = 0.9 / -0.3, which simplifies to |3 + 8x| = -3.
Upon isolating the absolute value, a crucial observation can be made by applying the fundamental principles of absolute values. The absolute value of any expression, irrespective of its complexity, is inherently non-negative. This is a direct consequence of the definition of absolute value, which measures the distance from zero and hence cannot yield a negative result. Therefore, any absolute value expression must be either zero or a positive number. In this context, the left side of the equation, |3 + 8x|, represents the absolute value of the expression 3 + 8x, which must be non-negative.
However, the right side of the equation is -3, a negative number. This creates a direct contradiction because we are equating a non-negative quantity (the absolute value expression) to a negative quantity. Such an equation cannot hold true for any value of x. Therefore, without any further calculation or exploration of potential cases, we can definitively conclude that the equation has no solution.
This understanding of the properties of absolute value is essential for solving such equations efficiently. It allows us to quickly identify situations where solutions are impossible, saving time and effort that might otherwise be spent on fruitless algebraic manipulations. Consequently, the equation -0.3|3 + 8x| = 0.9 has no solution, which aligns with the statement provided.
Determining the True Statement
After analyzing each equation, we can now determine which statement is true. Here's a summary of our findings:
- Equation 1: -3|2x + 1.2| = -1 has two solutions.
- Equation 2: 3.5|6x - 2| = 3.5 has two solutions.
- Equation 3: 5|-3.1x + 6.9| = -3.5 has no solution.
- Equation 4: -0.3|3 + 8x| = 0.9 has no solution.
Based on this analysis, the statement "The equation -0.3|3 + 8x| = 0.9 has no solution" is the only true statement. The other statements are false because they either claim the existence of solutions where none exist or misstate the number of solutions.
Conclusion
In conclusion, by meticulously analyzing each absolute value equation, we've identified that the statement "The equation -0.3|3 + 8x| = 0.9 has no solution" is the true statement. This process involved understanding the definition of absolute value, isolating the absolute value term, and considering the implications of equating an absolute value expression to positive, negative, or zero values. Mastering these steps is crucial for accurately solving absolute value equations and determining the validity of statements about their solutions. Understanding the properties of absolute values, especially their non-negativity, allows for quick identification of contradictions and ensures efficient problem-solving.
