Select the values that make the inequality 6, z, is less than, 126z<12 true. Then write an equivalent inequality, in terms of zz.

Question

(Numbers written in order from least to greatest going across.)
Answer
Attempt 1 out of 2

-8 -3 -1
1 2 3
5 7 12
Equivalent Inequality:

Answer 1

To solve the inequality \( 6z < 126z < 12 \), let's break it down into two parts.

1. First Inequality: \( 6z < 126z \)

   We can simplify this by subtracting \( 6z \) from both sides:

   \[ 0 < 120z \]

   Dividing both sides by \( 120 \) (which is positive):

   \[ 0 < z \quad \text{or} \quad z > 0 \]

2. 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

1. Selected value: \( 1 \)
2. Equivalent Inequality: \( 0 < z < \frac{2}{21} \)