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### Problem 17: 
A triangle has side lengths of \( 7x + 3 \) inches and \( 4x - 5 \) inches. If the perimeter of the triangle is 32 inches, which expression represents the length, in inches, of the third side of the triangle?

#### Solution:
The perimeter of a triangle is the sum of all three sides. Let the third side be \( y \). 
Thus, we can write:
\[
(7x + 3) + (4x - 5) + y = 32
\]
Simplifying the left-hand side:
\[
11x - 2 + y = 32
\]
Solving for \( y \):
\[
y = 32 - 11x + 2
\]
\[
y = 34 - 11x
\]

So, the expression for the third side is \( 34 - 11x \).

Answer: \( \boxed{34 - 11x} \).

---

### Problem 18: 
Which two expressions represent rational numbers?

Expressions given are:
(A) \( \frac{7}{9} + \sqrt{5} \)
(B) \( \frac{\sqrt{49}}{7} \)
(C) \( \sqrt{16} - \sqrt{81} \)
(D) \( \frac{\sqrt{3}}{2} \)

#### Solution:
- (A) \( \frac{7}{9} + \sqrt{5} \) is **not** a rational number because \( \sqrt{5} \) is irrational.
- (B) \( \frac{\sqrt{49}}{7} = \frac{7}{7} = 1 \), which is rational.
- (C) \( \sqrt{16} - \sqrt{81} = 4 - 9 = -5 \), which is rational.
- (D) \( \frac{\sqrt{3}}{2} \) is **not** rational because \( \sqrt{3} \) is irrational.

So, the rational expressions are **(B)** and **(C)**.

Answer: \( \boxed{B} \text{ and } \boxed{C} \).

---

### Problem 19: 
A box plot representing a data set is shown. The question asks for the interquartile range (IQR) of the data set.

#### Solution:
The interquartile range (IQR) is calculated as the difference between the third quartile (Q3) and the first quartile (Q1). Based on the box plot in the image:
- Q1 (first quartile) = 2
- Q3 (third quartile) = 8

Thus, the IQR is:
\[
\text{IQR} = Q3 - Q1 = 8 - 2 = 6
\]

Answer: \( \boxed{6} \).

---

### Problem 20:
A data set is represented by a plot where each "X" represents 1 data point. The question asks for the mean of the data.

#### Solution:
The data points shown are: \( 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 5 \).

To find the mean:
\[
\text{Mean} = \frac{\text{Sum of data points}}{\text{Number of data points}} = \frac{1 + 2(2) + 3(2) + 4(3) + 5(4)}{12}
\]
\[
\text{Sum} = 1 + 4 + 6 + 12 + 20 = 43
\]
\[
\text{Mean} = \frac{43}{12} \approx 3.58
\]

Answer: \( \boxed{3.58} \).

---

Let me know if you'd like further clarification or have additional questions! Here are five related questions to deepen your understanding:

1. How do you calculate the perimeter of any polygon?
2. What are the key properties of rational and irrational numbers?
3. How is the interquartile range helpful in identifying outliers in a data set?
4. Can you identify the difference between the mean and median in a dataset?
5. How would adding an outlier affect the mean of a dataset?

**Tip:** When simplifying square roots in rational expressions, always check if the square root results in a whole number!

A triangle has side lengths of 7x + 3 inches and 4x - 5 inches. If the perimeter of the triangle is 32 inches, which expression represents the length, in inches, of the third side of the triangle?

Solve for the third side of a triangle with side lengths 7x + 3 inches and 4x - 5 inches, and a perimeter of 32 inches. This algebra problem involves equation solving and basic perimeter concepts. Suitable for students in Grades 7-9.

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