To solve the inequality $x - \left(\frac{10}{x} - 1\right) \geq 4$, follow these steps:

1. Distribute the negative sign: $x - \frac{10}{x} + 1 \geq 4$

2. Move the constant on the right-hand side to the left: $x - \frac{10}{x} + 1 - 4 \geq 0$

3. Simplify the expression: $x - \frac{10}{x} - 3 \geq 0$

4. Combine like terms: $x - 3 - \frac{10}{x} \geq 0$

5. Multiply the entire inequality by $x$ (assuming $x > 0$, since $x < 0$ would reverse the inequality sign): $x(x - 3) - 10 \geq 0$

   This simplifies to: $x^2 - 3x - 10 \geq 0$

6. Solve the quadratic inequality: First, find the roots of the quadratic equation $x^2 - 3x - 10 = 0$.

   Using the quadratic formula: $x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-10)}}{2(1)}$ $x = \frac{3 \pm \sqrt{9 + 40}}{2}$ $x = \frac{3 \pm \sqrt{49}}{2}$ $x = \frac{3 \pm 7}{2}$

   So, the roots are: $x = \frac{10}{2} = 5 \quad \text{and} \quad x = \frac{-4}{2} = -2$

7. Analyze the intervals: The critical points divide the number line into three intervals:

   • $(-\infty, -2)$
   • $(-2, 5)$
   • $(5, \infty)$

   Check the sign of the quadratic expression in each interval.

   • For $x \in (-\infty, -2)$, the expression $x^2 - 3x - 10$ is