Pregunta
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\[ |c-6|=|c-5| \] The solution set is

Ask by Brewer Vaughn. in the United States
Feb 25,2025

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The solution is \( c = \frac{11}{2} \) or \( c = 5.5 \).

Solución

We want to solve the equation |c - 6| = |c - 5|. This equation says that the distance from c to 6 is equal to the distance from c to 5. One way to think about this is that c must be exactly halfway between 6 and 5. The midpoint of 6 and 5 is (6 + 5) / 2 = 5.5. Alternatively, we can solve algebraically: 1. Recognize that |c - 6| = |c - 5| implies either:   a) c - 6 = c - 5, which simplifies to -6 = -5 (this is false), or   b) c - 6 = -(c - 5). 2. Solve the second equation:   c - 6 = -c + 5   Add c to both sides: 2c - 6 = 5   Add 6 to both sides: 2c = 11   Divide by 2: c = 11/2 = 5.5 Therefore, the solution set is {11/2}.

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Beyond the Answer

When you have an equation like \( |c-6| = |c-5| \), it means the distances from \( c \) to 6 and from \( c \) to 5 are equal. This occurs when \( c \) is exactly midway between 5 and 6, which is 5.5. However, there are also two scenarios to consider: \( c - 6 = c - 5 \) (which doesn't provide new solutions) and \( c - 6 = -(c - 5) \). Solving the second yields \( c = 5.5 \). Thus, the solution set is \( \{5.5\} \). Another way to think about this is considering the geometric interpretation of the absolute value. The points represent locations on a number line. The value 5.5 is the balance point between 5 and 6, making it a sweet spot where both distances to 5 and 6 are the same. If you were to fold that number line at 5.5, both sides would align perfectly!

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