How do you solve the inequality: 7x > - 35?

To solve the inequality \( 7x > -35 \), follow these steps:

1. **Understand the inequality**: The inequality states that \( 7x \) is greater than \( -35 \). Our goal is to isolate \( x \) to determine the range of values it can take.

2. **Isolate \( x \)**:
- The term \( 7x \) means \( x \) is multiplied by 7. To isolate \( x \), divide both sides of the inequality by 7.
- Dividing by a positive number does not reverse the inequality sign. So:
\[
x > \frac{-35}{7}
\]

3. **Simplify the division**:
- Calculate \( \frac{-35}{7} \). This equals \( -5 \).
- Thus, the inequality simplifies to:
\[
x > -5
\]

4. **Interpret the solution**:
- The solution means that \( x \) can take any value greater than \( -5 \).
- In interval notation, this is represented as \( (-5, \infty) \), where \( \infty \) indicates no upper limit.

5. **Verify** (optional step):
- To check the solution, substitute a value greater than \( -5 \) (e.g., \( x = 0 \)) into the original inequality \( 7x > -35 \).
- Substituting \( x = 0 \):
\[
7(0) = 0 > -35 \, \text{(True)}
\]
- Similarly, substituting a value less than \( -5 \) (e.g., \( x = -6 \)) should make the inequality false:
\[
7(-6) = -42 \, \text{(which is not greater than } -35\text{)}.
\]

**Final Answer**: The solution is \( x > -5 \).