Which integer values of j would give the number `-37,129xx10^(j)` a value between `-110 and -1`?

Answer

To solve the problem of finding integer values of \( j \) such that the number \(-37.129 \times 10^j\) lies between \(-110\) and \(-1\), we can follow these steps:

### Step 1: Set up the inequality
We need to find the values of \( j \) such that:
\[ -110 < -37.129 \times 10^j < -1 \]

### Step 2: Remove the negative sign
Since all parts of the inequality are negative, we can multiply the entire inequality by \(-1\) (remember to reverse the inequality signs):
\[ 110 > 37.129 \times 10^j > 1 \]

### Step 3: Divide by 37.129
Next, we divide the entire inequality by \( 37.129 \):
\[ \frac{110}{37.129} > 10^j > \frac{1}{37.129} \]

Calculating the values:
- \( \frac{110}{37.129} \approx 2.96 \)
- \( \frac{1}{37.129} \approx 0.0269 \)

Thus, we have:
\[ 2.96 > 10^j > 0.0269 \]

### Step 4: Convert to logarithmic form
To solve for \( j \), we take the logarithm (base 10) of all parts of the inequality:
\[ \log_{10}(2.96) > j > \log_{10}(0.0269) \]

Calculating the logarithms:
- \( \log_{10}(2.96) \approx 0.471 \)
- \( \log_{10}(0.0269) \approx -1.571 \)

So we have:
\[ 0.471 > j > -1.571 \]

### Step 5: Identify integer values of \( j \)
The integer values of \( j \) that satisfy this inequality are:
- \( j = 0 \)
- \( j = -1 \)

### Conclusion
The integer values of \( j \) that give the number \(-37.129 \times 10^j\) a value between \(-110\) and \(-1\) are:
\[ j = 0 \text{ and } j = -1 \]

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