) If x is a nonzero real number, is x–2 always positive, always negative, or positive or negative depending on whether x is positive or negative? Explain your answer.
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To determine whether x-2 is always positive, always negative, or positive or negative depending on whether x is positive or negative, we need to analyze the expression further.
We know that x is a nonzero real number. So let's consider the two possible cases:
1. If x > 2:
In this case, x-2 will be greater than 0. Since x is greater than 2 and subtracting 2 from a number greater than 2 will yield a positive result. Therefore, x-2 will always be positive when x is positive.
2. If x < 2:
In this case, x-2 will be negative. Since x is less than 2, subtracting 2 from a number less than 2 will yield a negative result. Therefore, x-2 will always be negative when x is negative.
Based on the analysis of the cases, we can conclude that x-2 is positive or negative depending on whether x is positive or negative, respectively. It is not always positive or always negative.
In summary:
- If x > 2, x-2 is positive.
- If x < 2, x-2 is negative.
- If x = 2, x-2 is zero.
Hence, x-2 can have different signs depending on the value of x.