To solve the inequality
\[ \frac{49 - 7x}{x} > 7, \]
we begin by rearranging the inequality.
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Subtract 7 from both sides:
\[ \frac{49 - 7x}{x} - 7 > 0. \]
To combine the terms, we express 7 as a fraction:
\[ 7 = \frac{7x}{x}, \]
so we rewrite the inequality as:
\[ \frac{49 - 7x - 7x}{x} > 0. \]
This simplifies to:
\[ \frac{49 - 14x}{x} > 0. \]
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Set the numerator and denominator:
We need to solve the inequality:
\[ \frac{49 - 14x}{x} > 0. \]
The critical points occur when the numerator or denominator is zero:
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For the numerator \(49 - 14x = 0\): \[ 49 = 14x \implies x = \frac{49}{14} = \frac{7}{2} = 3.5. \]
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For the denominator \(x = 0\).
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Test intervals:
The critical points \(x = 0\) and \(x = 3.5\) divide the number line into intervals:
- \( (-\infty, 0) \)
- \( (0, 3.5) \)
- \( (3.5, \infty) \)
We need to test a point from each interval to see where the inequality holds true.
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Interval \( (-\infty, 0) \): Choose \(x = -1\): \[ \frac{49 - 14(-1)}{-1} = \frac{49 + 14}{-1} = \frac{63}{-1} = -63 \quad \text{(not greater than 0)}. \]
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Interval \( (0, 3.5) \): Choose \(x = 1\): \[ \frac{49 - 14(1)}{1} = \frac{49 - 14}{1} = \frac{35}{1} = 35 \quad \text{(greater than 0)}. \]
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Interval \( (3.5, \infty) \): Choose \(x = 4\): \[ \frac{49 - 14(4)}{4} = \frac{49 - 56}{4} = \frac{-7}{4} \quad \text{(not greater than 0)}. \]
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Conclusion:
The solution to the inequality is the interval where the expression is greater than 0:
\( \boxed{(0, 3.5)}. \)
Now, filling this into the answers:
Part B: The solutions are the values of \(x\) in the interval: \[ (0, 3.5). \]
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