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The inequality provided is:

\[
x^2 - 7x + 10 \geq 0
\]

To solve this inequality, we’ll proceed with the following steps:

### Step 1: Factor the Quadratic Expression
We want to factor \( x^2 - 7x + 10 \):
\[
x^2 - 7x + 10 = (x - 5)(x - 2)
\]

### Step 2: Set Each Factor to Zero
The roots of the equation \( (x - 5)(x - 2) = 0 \) are:
\[
x = 5 \quad \text{and} \quad x = 2
\]

These values split the number line into three intervals: \( (-\infty, 2) \), \( (2, 5) \), and \( (5, \infty) \).

### Step 3: Determine the Sign of Each Interval
To determine where \( (x - 5)(x - 2) \geq 0 \), we test each interval:

1. **Interval \( (-\infty, 2) \):** Choose \( x = 1 \):
   \[
   (1 - 5)(1 - 2) = (-4)(-1) = 4 \quad (\text{positive})
   \]

2. **Interval \( (2, 5) \):** Choose \( x = 3 \):
   \[
   (3 - 5)(3 - 2) = (-2)(1) = -2 \quad (\text{negative})
   \]

3. **Interval \( (5, \infty) \):** Choose \( x = 6 \):
   \[
   (6 - 5)(6 - 2) = (1)(4) = 4 \quad (\text{positive})
   \]

### Step 4: Combine the Results
Since we want the expression to be non-negative (\( \geq 0 \)), the solution is the union of the intervals where the expression is positive or zero:

\[
x \in (-\infty, 2] \cup [5, \infty)
\]

### Final Answer:
\[
x \in (-\infty, 2] \cup [5, \infty)
\]

Would you like a detailed explanation on any part? 

Here are some related questions for further practice:

1. How do you solve a similar quadratic inequality with a negative leading coefficient?
2. What if the inequality were strict (\( > 0 \)) rather than inclusive (\( \geq 0 \))?
3. How do we handle inequalities when the quadratic does not factor neatly?
4. How would this solution change if the quadratic were \( x^2 + 7x + 10 \geq 0 \)?
5. How can we verify the solution by graphing the function \( y = x^2 - 7x + 10 \)?

**Tip:** When solving quadratic inequalities, always check each interval carefully, as this ensures accurate determination of positive or negative signs.

Solve the inequality: x^2 - 7x + 10 ≥ 0

Learn how to solve the quadratic inequality x^2 - 7x + 10 ≥ 0. This solution covers key algebraic concepts, including factoring quadratic expressions, analyzing intervals, and determining where the expression is non-negative. Ideal for high school students studying quadratic inequalities.

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