https://file.aiquickdraw.com/mathbot/file/1730147447728k9bmcr0v.jpg

Which point is a solution to the simultaneous inequality y < 1/2 xand y > 1/3 x + 3?  A. (10, 5)   B. (3, 6)  C. (2, 2)  D. (25, 12) 

To determine which point is a solution to the simultaneous inequalities:

1. \( y < \frac{1}{2}x \)
2. \( y > \frac{1}{3}x + 3 \)

we'll substitute each point into these inequalities and see if both conditions are satisfied.

### Checking Points

#### Option A: (10, 5)
1. \( y = 5 < \frac{1}{2} \times 10 = 5 \) (False, since \(5\) is not less than \(5\))
2. \( y = 5 > \frac{1}{3} \times 10 + 3 = \frac{10}{3} + 3 = \frac{19}{3} \approx 6.33 \) (False, since \(5\) is not greater than \(6.33\))

**Conclusion**: Not a solution.

#### Option B: (3, 6)
1. \( y = 6 < \frac{1}{2} \times 3 = 1.5 \) (False, since \(6\) is not less than \(1.5\))
2. \( y = 6 > \frac{1}{3} \times 3 + 3 = 1 + 3 = 4 \) (True)

**Conclusion**: Not a solution.

#### Option C: (2, 2)
1. \( y = 2 < \frac{1}{2} \times 2 = 1 \) (False, since \(2\) is not less than \(1\))
2. \( y = 2 > \frac{1}{3} \times 2 + 3 = \frac{2}{3} + 3 = \frac{11}{3} \approx 3.67 \) (False)

**Conclusion**: Not a solution.

#### Option D: (25, 12)
1. \( y = 12 < \frac{1}{2} \times 25 = 12.5 \) (True)
2. \( y = 12 > \frac{1}{3} \times 25 + 3 = \frac{25}{3} + 3 = \frac{34}{3} \approx 11.33 \) (True)

**Conclusion**: This point **satisfies both inequalities**.

### Answer
The point that satisfies both inequalities is **D. (25, 12)**.

Would you like further clarification or additional details?

---

Here are 5 related questions for deeper understanding:

1. How do you graph linear inequalities on a coordinate plane?
2. What is the process for finding the solution region for two inequalities?
3. How can you check if a point lies in a shaded region on a graph?
4. What does the boundary line of an inequality represent in a graph?
5. How can simultaneous inequalities be used in real-world applications?

**Tip:** When checking inequalities, always evaluate each inequality independently to ensure each condition is satisfied.

Which point is a solution to the simultaneous inequality y < 1/2 x and y > 1/3 x + 3?  A. (10, 5)   B. (3, 6)  C. (2, 2)  D. (25, 12)

Determine which point satisfies the simultaneous linear inequalities y < 1/2 x and y > 1/3 x + 3. This problem explores algebraic and coordinate geometry concepts suitable for Grades 9-10, involving the substitution of points to verify solutions to inequalities.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation