We need to solve four problems: Problem 8: Determine the correct logical expression representing "The sun is shining if and only if the weather is warm," given the statements $p$: "The weather is warm" and $q$: "The sun is shining." Problem 9: Identify the set property illustrated by the equation $P \cap (Q \cup R) = (P \cap Q) \cup (P \cap R)$. Problem 10: Simplify the expression $3\frac{1}{3} \times 4\frac{1}{2} \div 2\frac{1}{2}$. Problem 11: Given that $8^{x+1} = \frac{1}{4}$, find the value of $x$.

1. Problem Description

We need to solve four problems:

Problem 8: Determine the correct logical expression representing "The sun is shining if and only if the weather is warm," given the statements $p$: "The weather is warm" and $q$: "The sun is shining."

Problem 9: Identify the set property illustrated by the equation $P \cap (Q \cup R) = (P \cap Q) \cup (P \cap R)$.

Problem 10: Simplify the expression $3\frac{1}{3} \times 4\frac{1}{2} \div 2\frac{1}{2}$.

Problem 11: Given that $8^{x+1} = \frac{1}{4}$, find the value of $x$.

2. Solution Steps

Problem 8:

The statement "The sun is shining if and only if the weather is warm" translates to $q \Leftrightarrow p$. This means $q$ is true if and only if $p$ is true.

Problem 9:

The equation $P \cap (Q \cup R) = (P \cap Q) \cup (P \cap R)$ demonstrates the distributive property of intersection over union.

Problem 10:

First, convert the mixed numbers to improper fractions:

$3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$

$4\frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}$

$2\frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2}$

Now, substitute these values into the expression:

$\frac{10}{3} \times \frac{9}{2} \div \frac{5}{2} = \frac{10}{3} \times \frac{9}{2} \times \frac{2}{5}$

$= \frac{10 \times 9 \times 2}{3 \times 2 \times 5} = \frac{180}{30} = 6$

Problem 11:

Given $8^{x+1} = \frac{1}{4}$, we want to find $x$.

Rewrite the equation using powers of 2:

$(2^3)^{x+1} = 2^{-2}$

$2^{3(x+1)} = 2^{-2}$

Equate the exponents:

$3(x+1) = -2$

$3x + 3 = -2$

$3x = -5$

$x = -\frac{5}{3}$

3. Final Answer

Problem 8: D. $q \Leftrightarrow p$

Problem 9: B. Distributive

Problem 10: C. 6

Problem 11: A. $-\frac{5}{3}$