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"Check Your Understanding" Section

Example 1: Writing Algebraic Expressions

1. The product of 12 and the sum of a number and -3: $12 \cdot (x + (-3)) \quad \text{or} \quad 12(x - 3)$

2. The difference between the product of 4 and a number and the square of the number: $4x - x^2$

Example 2: Writing Verbal Sentences for Equations

3. $x^2 - 9 = 27$: The square of a number minus 9 equals 27.

4. $x + 8 = 16$: A number increased by 8 equals 16.

Example 3: Properties of Operations

5. $(8x - 3) + 12 = (8x - 3) + 12$:
   Illustrates the Associative Property of Addition.

6. If $a = -3$ and $-3 = d$, then $a = d$:
   Illustrates the Transitive Property of Equality.

Examples 4-5: Solving Equations

7. Solve $2 = 19 - 34$: Simplify to find $x$:
   $x = -16$

8. Solve $x + 13 = 7$: Subtract 13 from both sides:
   $x = -6$

9. Solve $5x - 3 = -33$: Add 3 and divide by 5:
   $x = -6$

Example 6: Solve for a Variable

10. $8r - 5q = 3,$ solve for $q$:
    Rearrange:
    $q = \frac{8r - 3}{5}$

Example 7: Multiple Choice

11. If $\frac{5}{y} + 8 = 7$, solve for $y$:
    Subtract 8 from 7, then solve:
    $y = -5$ Correct answer: C

"Practice and Problem Solving" Section

Example 1: Writing Algebraic Expressions

22. The difference between the product of four and a number, and 6:
    $4x - 6$

23. The product of the square of a number and 8:
    $8x^2$

24. Fifteen less than the cube of a number:
    $x^3 - 15$

25. Five more than the quotient of a number and 4:
    $\frac{x}{4} + 5$

Would you like a detailed explanation for any specific problem or additional examples?

Expanding Questions:

1. How does one identify the correct property for Example 3?
2. What are the steps to isolate variables in linear equations like Example 6?
3. Why does the solution in Example 7 require the reciprocal when solving?
4. What are key differences between expressions and equations in Example 1?
5. Can you apply similar properties to inequalities?

Tip: When solving equations, always double-check your steps by substituting your solution back into the original equation.