Question

1.
stephanie is solving the equation $x^2 - 12 = 7x - 8$. her first step is shown below.
given: $x^2 - 12 = 7x - 8$
step 1: $x^2 - 4 = 7x$
which property justifies her first step?
(1) associative property \t\t(3) distributive property
(2) commutative property \t\t(4) addition property of equality

2.
at berkeley central high school, a survey was conducted to see if students preferred cheeseburgers, pizza, or hot dogs for lunch. the results of this survey are shown in the table below.

	cheeseburgers	pizza	hot dogs
males	36	30	34

based on this survey, what percent of the students preferred pizza?
(1) 30 \t\t\t(3) 44
(2) 37 \t\t\t(4) 74

3.
if $f(x) = \frac{3x + 4}{2}$, then $f(8)$ is
(1) 21 \t\t\t(3) 14
(2) 16 \t\t\t(4) 4

4.
the expression $36x^2 - 9$ is equivalent to
(1) $(6x - 3)^2$ \t\t\t(3) $(6x + 3)(6x - 3)$
(2) $(18x - 4.5)^2$ \t\t\t(4) $(18x + 4.5)(18x - 4.5)$

Explanation:

Question 1

Step1: Recall properties of equality

The addition property of equality states that if we add the same number to both sides of an equation, the equation remains true.

Step2: Analyze Stephanie's step

Given \( x^2 - 12=7x - 8 \), she got \( x^2-4 = 7x \). To get from the first equation to the second, she added 8 to both sides: \( (x^2 - 12)+ 8=(7x - 8)+ 8\), which simplifies to \( x^2-4 = 7x \). This is using the addition property of equality. The associative property is for grouping, commutative for order, and distributive for multiplication over addition, so they don't apply here.

Step1: Find total number of students

First, calculate the total number of students. For females: \( 32 + 44+24=100 \). For males: \( 36 + 30+34 = 100 \). So total students \(=100 + 100=200 \).

Step2: Find number of students who prefer pizza

Number of students who prefer pizza: females who prefer pizza (\( 44 \)) plus males who prefer pizza (\( 30 \)) \(=44 + 30 = 74 \).

Step3: Calculate the percentage

Percentage \(=\frac{\text{Number of pizza - lovers}}{\text{Total number of students}}\times100=\frac{74}{200}\times 100 = 37\%\)

Step1: Substitute \( x = 8 \) into the function

Given \( f(x)=\frac{3x + 4}{2} \), substitute \( x = 8 \): \( f(8)=\frac{3(8)+4}{2} \)

Step2: Simplify the numerator

Calculate the numerator: \( 3\times8+4=24 + 4=28 \)

Step3: Divide by 2

Then \( f(8)=\frac{28}{2}=14 \)

Answer:

(4) addition property of equality

Question 2