Final answer:
The given expression (|x|/x) simplifies to -1 when x is negative and to 1 when x is positive. For x = -8, -5, the expression is -1, and for x = 1, 7, 128, it is 1.
Step-by-step explanation:
The expression \(|x|/x\) evaluates the sign of a real number x. When x is negative, |x| equals -x because the absolute value turns the negative x into a positive value, but since x itself is negative, the expression simplifies to \((-x)/x\), which equals -1. When x is positive, |x| equals x, and therefore \(x/x\) simplifies to 1. So, the expression evaluates to -1 for negative values of x, and 1 for positive values of x.
For the values given in the question, we can evaluate the expression as follows:
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- For x = -8: |x|/x = |-8|/-8 = 8/-8 = -1
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- For x = -5: |x|/x = |-5|/-5 = 5/-5 = -1
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- For x = 1: |x|/x = |1|/1 = 1/1 = 1
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- For x = 7: |x|/x = |7|/7 = 7/7 = 1
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- For x = 128: |x|/x = |128|/128 = 128/128 = 1
The expression without the absolute value symbol for negative values of x is \(-1\), since for every x < 0, the expression simplifies to \(-1\).