When grappling with absolute value equations, determining the nature and number of solutions can be a fascinating mathematical journey. This article delves into the intricacies of absolute value equations, carefully examining several statements to pinpoint the one that holds true. We'll dissect each equation, step by step, to reveal whether it boasts no solution, a single solution, or a pair of solutions. This exploration will not only sharpen your understanding of absolute value equations but also enhance your problem-solving prowess in the realm of algebra.
Understanding Absolute Value Equations
Before we embark on our quest to identify the true statement, let's fortify our understanding of absolute value equations. Absolute value, denoted by vertical bars | |, represents the distance of a number from zero on the number line. Consequently, the absolute value of a number is always non-negative. For instance, |3| = 3 and |-3| = 3. This fundamental concept underpins the solutions to absolute value equations.
An absolute value equation typically takes the form |ax + b| = c, where a, b, and c are constants. Solving such an equation involves considering two distinct cases: ax + b = c and ax + b = -c. This stems from the fact that both a positive number and its negative counterpart have the same absolute value. However, the existence and nature of solutions hinge critically on the value of 'c'. If 'c' is negative, the equation has no solution, as the absolute value cannot be negative. If 'c' is zero, there is exactly one solution. If 'c' is positive, the equation generally has two solutions.
Now, with this bedrock of knowledge, let's dissect the given statements and unveil the truth.
Statement 1: The equation -3|2x + 1.2| = -1 has no solution.
To assess the veracity of this statement, we must meticulously analyze the equation -3|2x + 1.2| = -1. Our initial step involves isolating the absolute value term. Dividing both sides of the equation by -3, we arrive at |2x + 1.2| = 1/3. Now, we have an absolute value expression set equal to a positive number. This implies that the equation potentially has solutions. To find these solutions, we must consider the two cases inherent in absolute value equations.
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Case 1: 2x + 1.2 = 1/3
Subtracting 1.2 from both sides yields 2x = 1/3 - 1.2. Converting 1.2 to a fraction, we have 1.2 = 6/5. Thus, 2x = 1/3 - 6/5. Finding a common denominator, we get 2x = 5/15 - 18/15, which simplifies to 2x = -13/15. Dividing both sides by 2, we find x = -13/30.
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Case 2: 2x + 1.2 = -1/3
Subtracting 1.2 (or 6/5) from both sides gives 2x = -1/3 - 6/5. Again, finding a common denominator, we have 2x = -5/15 - 18/15, resulting in 2x = -23/15. Dividing both sides by 2, we obtain x = -23/30.
We have unearthed two distinct solutions: x = -13/30 and x = -23/30. This definitively demonstrates that the statement "The equation -3|2x + 1.2| = -1 has no solution" is false. The equation, in fact, possesses two solutions.
Statement 2: The equation 3.5|6x - 2| = 3.5 has one solution.
Let's delve into the second statement, which asserts that the equation 3.5|6x - 2| = 3.5 has one solution. Our initial move is to isolate the absolute value term. Dividing both sides of the equation by 3.5, we simplify the equation to |6x - 2| = 1. Now, we have a standard absolute value equation ripe for analysis.
To solve |6x - 2| = 1, we must consider the two cases that arise from the definition of absolute value:
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Case 1: 6x - 2 = 1
Adding 2 to both sides, we get 6x = 3. Dividing both sides by 6, we find x = 1/2.
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Case 2: 6x - 2 = -1
Adding 2 to both sides, we obtain 6x = 1. Dividing both sides by 6, we find x = 1/6.
We have discovered two distinct solutions: x = 1/2 and x = 1/6. This unequivocally demonstrates that the statement "The equation 3.5|6x - 2| = 3.5 has one solution" is false. The equation boasts two solutions, not one.
Statement 3: The equation 5|-3.1x + 6.9| = -3.5 has two solutions.
The third statement posits that the equation 5|-3.1x + 6.9| = -3.5 has two solutions. Our familiar first step is to isolate the absolute value term. Dividing both sides of the equation by 5, we arrive at |-3.1x + 6.9| = -3.5/5, which simplifies to |-3.1x + 6.9| = -0.7. Here, we encounter a crucial observation.
We have an absolute value expression, |-3.1x + 6.9|, set equal to a negative number, -0.7. This is a mathematical impossibility. The absolute value of any expression, by definition, is always non-negative. Therefore, no value of x can satisfy this equation. Consequently, the equation 5|-3.1x + 6.9| = -3.5 has no solution.
This definitively proves that the statement "The equation 5|-3.1x + 6.9| = -3.5 has two solutions" is false. The equation has no solutions.
Statement 4: The equation -0.3|3 + 8x| = 0.9 has no solution.
Finally, let's scrutinize the fourth statement, which asserts that the equation -0.3|3 + 8x| = 0.9 has no solution. As before, our initial move is to isolate the absolute value term. Dividing both sides of the equation by -0.3, we obtain |3 + 8x| = -3. Again, we arrive at a pivotal observation.
We have an absolute value expression, |3 + 8x|, equated to a negative number, -3. As we established earlier, the absolute value of any expression can never be negative. Therefore, no value of x can satisfy this equation. Consequently, the equation -0.3|3 + 8x| = 0.9 indeed has no solution.
This definitively validates that the statement "The equation -0.3|3 + 8x| = 0.9 has no solution" is true. This is the statement we have been searching for.
Conclusion: The True Statement Revealed
Through our rigorous analysis of each statement, we have unearthed the truth. The statement "The equation -0.3|3 + 8x| = 0.9 has no solution" is the true statement among the options presented. This statement holds true because isolating the absolute value term reveals that it is equated to a negative number, a mathematical impossibility. Absolute value, by its very definition, can never be negative.
This exploration has not only pinpointed the true statement but also reinforced our understanding of absolute value equations. Remember, when tackling absolute value equations, always isolate the absolute value term first. Then, carefully consider the sign of the constant on the other side of the equation. If the constant is negative, the equation has no solution. If the constant is non-negative, proceed to solve the two cases that arise from the definition of absolute value.
By mastering these principles, you will be well-equipped to navigate the world of absolute value equations with confidence and accuracy.
By dissecting these equations and applying the fundamental principles of absolute value, we've not only answered the question but also deepened our understanding of this essential mathematical concept. Understanding these nuances allows for accurate problem-solving and a stronger grasp of algebraic principles.
