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### Solution for Selected Problems from Pre-AP Geometry and Pre-AP Algebra 2 Summer Assignment

Let's solve a few of the problems from the provided summer assignment document.

#### Problem 1: Simplify \(\left( 64 - 5 \cdot 3^2 \right) + 7 \cdot 2^3 + \frac{18}{2^2} - 5^2\)

First, we simplify inside the parentheses and exponents:
\[64 - 5 \cdot 9 + 7 \cdot 8 + \frac{18}{4} - 25\]

Calculate each term step-by-step:
\[64 - 45 + 56 + 4.5 - 25\]

Combine the terms:
\[64 - 45 = 19\]
\[19 + 56 = 75\]
\[75 + 4.5 = 79.5\]
\[79.5 - 25 = 54.5\]

So, the simplified expression is:
\[54.5\]

#### Problem 2: Evaluate \(\frac{4x^2 - 3y + z}{2xy - 3z + 16} \text{ when } x = 2, y = -5, z = 3\)

First, substitute the values \(x = 2\), \(y = -5\), and \(z = 3\) into the expression:
\[\frac{4(2)^2 - 3(-5) + 3}{2(2)(-5) - 3(3) + 16}\]

Calculate the numerator:
\[4(2)^2 = 16\]
\[-3(-5) = 15\]
\[16 + 15 + 3 = 34\]

Calculate the denominator:
\[2(2)(-5) = -20\]
\[-3(3) = -9\]
\[-20 - 9 + 16 = -13\]

So the expression becomes:
\[\frac{34}{-13} = -\frac{34}{13}\]

#### Problem 3: Is \(x = -4\) a solution to the inequality \(10 \leq 5x^2 + x - 25\)?

First, substitute \(x = -4\) into the inequality:
\[10 \leq 5(-4)^2 + (-4) - 25\]

Calculate the right side:
\[5(-4)^2 = 5(16) = 80\]
\[-4 - 25 = -29\]
\[80 - 29 = 51\]

So the inequality becomes:
\[10 \leq 51\]

This is true, so \(x = -4\) is a solution to the inequality.

#### Problem 4: Evaluate \(x - zy + z^2 - y^2\) when \(x = 2\), \(y = -4\), and \(z = -7\)

First, substitute the values \(x = 2\), \(y = -4\), and \(z = -7\):
\[2 - (-7)(-4) + (-7)^2 - (-4)^2\]

Calculate each term:
\[-7(-4) = 28\]
\[-7^2 = 49\]
\[-4^2 = 16\]

So, the expression becomes:
\[2 - 28 + 49 - 16\]

Combine the terms:
\[2 - 28 = -26\]
\[-26 + 49 = 23\]
\[23 - 16 = 7\]

Thus, the value is:
\[7\]

### Next Steps

Would you like detailed solutions for more problems or explanations of specific steps? Here are some related questions you might be interested in:

1. What are the properties of exponents used in Problem 41?
2. Can you explain the process of graphing a linear equation?
3. How do you solve a system of linear equations by substitution?
4. What is the point-slope form of a line equation and how is it derived?
5. How do you determine if two lines are parallel or perpendicular?
6. How do you solve a quadratic equation by completing the square?
7. What are perfect square trinomials and how are they identified?
8. How do you factor polynomials and what methods are commonly used?

#### Tip
When simplifying expressions, always follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).

Explore detailed solutions to selected problems from the Pre-AP Geometry and Pre-AP Algebra 2 summer assignment. Includes step-by-step solutions for simplifying expressions, evaluating algebraic expressions, and solving inequalities. Suitable for high school students.

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