https://file.aiquickdraw.com/mathbot/file/17258241557469okwd737.jpg

The image contains a mathematical inequality:  
\[
x^2 - 7x + 12 \leq 0
\]

The task is to determine which of the following values are solutions to this inequality:  
1. \( x = 1 \)  
2. \( x = 3 \)  
3. \( x = 5 \)  
4. \( x = 7 \)

Let's solve the inequality to identify the correct values.

### Step 1: Solve \(x^2 - 7x + 12 = 0\)  
Factorizing the quadratic equation:
\[
x^2 - 7x + 12 = (x - 3)(x - 4) = 0
\]
The solutions to the equation are \(x = 3\) and \(x = 4\).

### Step 2: Analyze the inequality
The inequality is \( (x - 3)(x - 4) \leq 0 \), which is true when \(x\) lies between 3 and 4, inclusive. Therefore, the solution set is \( 3 \leq x \leq 4 \).

### Step 3: Test the given values:
- \( x = 1 \): Substituting in the inequality, \( (1 - 3)(1 - 4) = (-2)(-3) = 6 > 0 \). Not a solution.
- \( x = 3 \): \( (3 - 3)(3 - 4) = 0 \times (-1) = 0 \). A solution.
- \( x = 5 \): \( (5 - 3)(5 - 4) = (2)(1) = 2 > 0 \). Not a solution.
- \( x = 7 \): \( (7 - 3)(7 - 4) = (4)(3) = 12 > 0 \). Not a solution.

### Conclusion:
The only solution among the given options is \( x = 3 \).

Would you like further details or have any other questions?

Here are 5 related questions you could explore:
1. How do you factor quadratic equations?
2. What is the process to solve quadratic inequalities?
3. Can you explain the difference between strict and non-strict inequalities?
4. How do you graph the solution of a quadratic inequality?
5. What are some other methods to solve quadratic inequalities, apart from factoring?

**Tip:** When solving inequalities, always remember to check for boundary values and test intervals between solutions!

Which of the following values are solutions to the inequality: x^2 - 7x + 12 ≤ 0?
1. x = 1
2. x = 3
3. x = 5
4. x = 7

Discover how to solve the quadratic inequality x^2 - 7x + 12 ≤ 0 and test specific values for solutions. This problem involves key algebra concepts such as factoring quadratics and analyzing inequality intervals. Ideal for Grades 9-12.

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