To solve the inequality \(3x+1 > \sqrt{x} - 5\) using the fact that \(f(x) = \sqrt{x}\) is increasing over its domain, follow these steps:
1. **Isolate the Square Root Term**:
\(3x + 1 > \sqrt{x} - 5\)
Add 5 to both sides:
\(3x + 6 > \sqrt{x}\)
2. **Square Both Sides**:
Since \(\sqrt{x}\) is increasing, squaring the inequality is allowed without changing the direction of the inequality.
\((3x + 6)^2 > x\)
3. **Expand and Simplify**:
\((3x + 6)^2 = 9x^2 + 36x + 36\)
The inequality becomes:
\(9x^2 + 36x + 36 > x\)
4. **Rearrange**:
\(9x^2 + 36x + 36 - x > 0\)
\(9x^2 + 35x + 36 > 0\)
5. **Solve the Quadratic Inequality**:
You can solve the quadratic inequality using various methods like factoring, the quadratic formula, or plotting the graph of the quadratic function.
6. **Verify Solutions**:
After finding the solutions to the inequality, plug them back into the original inequality to ensure they satisfy the initial condition.
By following these steps and utilizing the fact that \(\sqrt{x}\) is increasing, you can effectively solve the given inequality.
