1

*x* − 1

 ≥ 

*x* − 1

*x* + 5

Let's solve the inequality:

\[
x - 1 \geq \frac{x - 1}{x + 5}
\]

### Step 1: Analyze the inequality
The inequality has a fraction, so let's first eliminate the fraction by multiplying both sides by \(x + 5\). However, we need to be careful with the sign of \(x + 5\) since it will affect the direction of the inequality.

The inequality becomes:

\[
(x - 1)(x + 5) \geq x - 1
\]

### Step 2: Simplify the inequality
Now, expand the left side of the inequality:

\[
x^2 + 5x - x - 5 \geq x - 1
\]

Simplify this further:

\[
x^2 + 4x - 5 \geq x - 1
\]

### Step 3: Move all terms to one side
Subtract \(x - 1\) from both sides:

\[
x^2 + 4x - 5 - x + 1 \geq 0
\]

Simplify further:

\[
x^2 + 3x - 4 \geq 0
\]

### Step 4: Factor the quadratic
Factor the quadratic expression:

\[
(x + 4)(x - 1) \geq 0
\]

### Step 5: Determine the intervals
The solutions to the equation \((x + 4)(x - 1) = 0\) are \(x = -4\) and \(x = 1\).

We now determine where the product \((x + 4)(x - 1)\) is non-negative by testing the sign of the expression in the intervals determined by these roots:

1. \(x < -4\)
2. \(-4 < x < 1\)
3. \(x > 1\)

### Step 6: Test the signs in each interval
- For \(x < -4\) (e.g., \(x = -5\)): \((x + 4)\) is negative, \((x - 1)\) is negative, and their product is positive.
- For \(-4 < x < 1\) (e.g., \(x = 0\)): \((x + 4)\) is positive, \((x - 1)\) is negative, and their product is negative.
- For \(x > 1\) (e.g., \(x = 2\)): \((x + 4)\) is positive, \((x - 1)\) is positive, and their product is positive.

### Step 7: Include boundary points
The inequality \((x + 4)(x - 1) \geq 0\) is satisfied when the product is positive or zero. Therefore, the solution is:

\[
(-\infty, -4] \cup [1, \infty)
\]

### Step 8: Consider domain restrictions
The original inequality involves the term \(x + 5\) in the denominator, so we need to ensure that \(x + 5 \neq 0\), which implies \(x \neq -5\). However, \(x = -5\) is not within our solution set, so the solution remains:

\[
(-\infty, -4] \cup [1, \infty)
\]

Would you like a number line representation or further explanation?

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**Relative Questions:**

1. Why does the sign of the factor \((x + 5)\) matter when multiplying both sides of the inequality?
2. What is the significance of including the boundary points \(-4\) and \(1\) in the solution?
3. How does factoring a quadratic expression help in solving inequalities?
4. What would happen if the inequality was strict (i.e., \(>\) instead of \(\geq\))?
5. How can domain restrictions change the solution of an inequality?

**Tip:** Always test the boundary points separately when solving inequalities to determine if they should be included in the solution set.

Learn how to solve the inequality x - 1 >= (x - 1) / (x + 5) with a step-by-step solution. This problem covers key algebraic concepts including inequalities, quadratic equations, and factoring. Suitable for students in Grades 10-12.

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