To solve the inequality \( 6z < 126z < 12 \), let's break it down into two parts.
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First Inequality: \( 6z < 126z \)
We can simplify this by subtracting \( 6z \) from both sides:
\[ 0 < 120z \]
Dividing both sides by \( 120 \) (which is positive):
\[ 0 < z \quad \text{or} \quad z > 0 \]
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Second Inequality: \( 126z < 12 \)
We can simplify this by dividing both sides by \( 126 \):
\[ z < \frac{12}{126} \]
Simplifying \( \frac{12}{126} \):
\[ \frac{12}{126} = \frac{2}{21} \]
Thus, the second part gives us:
\[ z < \frac{2}{21} \]
Now, combining both results, we can write the compound inequality as:
\[ 0 < z < \frac{2}{21} \]
In interval notation, this corresponds to the interval \( (0, \frac{2}{21}) \).
Select the Values
From the provided options, we check which values fall within the interval \( (0, \frac{2}{21}) \). Since \( \frac{2}{21} \approx 0.0952 \), we see that the only number in the provided options falling within this interval is \( 1 \).
Equivalent Inequality
The equivalent inequality in terms of \( z \) can therefore be written as:
\[ 0 < z < \frac{2}{21} \]
Answers
- Selected value: \( 1 \)
- Equivalent Inequality: \( 0 < z < \frac{2}{21} \)