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

Learn how to solve the quadratic inequality x - (10/x - 1) ≥ 4 with a step-by-step solution. This problem involves key algebraic concepts such as simplifying rational expressions and applying the quadratic formula. Suitable for students in Grades 9-11.

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