Question

The difference of a number t and 7 is greater than 10 and less than 20

Answer 1

To express the statement "The difference of a number \( t \) and 7 is greater than 10 and less than 20" mathematically, we can write:

\[
10 < |t - 7| < 20
\]

This translates to two separate inequalities:

1. \( |t - 7| > 10 \)
2. \( |t - 7| < 20 \)

Now let's analyze these inequalities.

### From \( |t - 7| > 10 \):

This means that the expression \( t - 7 \) is either greater than 10 or less than -10:

1. \( t - 7 > 10 \) leads to \( t > 17 \)
2. \( t - 7 < -10 \) leads to \( t < -3 \)

So, from this part, we have:

\[
t > 17 \quad \text{or} \quad t < -3
\]

### From \( |t - 7| < 20 \):

This means that the expression \( t - 7 \) is between -20 and 20:

- \( -20 < t - 7 < 20 \)

Adding 7 to each part of the inequality:

- \( -20 + 7 < t < 20 + 7 \)
- \( -13 < t < 27 \)

### Combining the results:

Now, we need to combine the inequalities:

1. From \( |t - 7| > 10 \), we have \( t > 17 \) or \( t < -3 \).
2. From \( |t - 7| < 20 \), we have \( -13 < t < 27 \).

### Finding overlapping intervals:

- For \( t < -3 \), the overlap with \( -13 < t < 27 \) gives us \( -13 < t < -3 \).
- For \( t > 17 \), the overlap with \( -13 < t < 27 \) gives us \( 17 < t < 27 \).

### Final result:

Putting this together, the final solution is:

\[
-13 < t < -3 \quad \text{or} \quad 17 < t < 27
\]