Question

exercise

1. alice is selling lemonade and fresh-squeezed orange juice at a booth. she sells cups of lemonade for \\$1 each and cups of orange juice for \\$3 each. write an expression that represents the amount of money that alice earns selling drinks.

1. evaluate \\(2x - 4\\) for \\(x = 9\\).

1. evaluate \\(-b + 8\\) for \\(b = 3\\).

1. find the value of the expression \\(\frac{1}{2}z + 3\\) if \\(x = 12\\).

1. for the equation \\(y = 3x + 4\\), find the value of \\(y\\) if \\(x = 2\\).

1. the expression \\(\\$25 + \\$10h\\) gives the amount of money that stephanie earns per day, where \\(h\\) is the number of hours she works. how much does she earn if she works 4 hours?

1. simplify each of the following expressions:

a. \\(3(x + 2) + 4(x - 2)\\)
b. \\(2(x + y) - 3(x + y)\\)
c. \\(-(y + 8) + 4(6 - y)\\)
d. \\(2(4x + 3) - 3(x + 1)\\)

Explanation:

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<reasoning>

Formulate expression for drink sales

\[

$$\begin{aligned} &\text{Let } L = \text{number of cups of lemonade sold}\\ &\text{Let } O = \text{number of cups of orange juice sold}\\ &\text{Total earnings} = 1 \cdot L + 3 \cdot O = L + 3O \end{aligned}$$

\]

Evaluate expressions for questions 2 to 6

\[

$$\begin{aligned} &\text{Q2: } 2x - 4 \text{ for } x = 9 \implies 2(9) - 4 = 18 - 4 = 14\\ &\text{Q3: } -b + 8 \text{ for } b = 3 \implies -(3) + 8 = 5\\ &\text{Q4: } \frac{1}{2}z + 3 \text{ for } z = 12 \implies \frac{1}{2}(12) + 3 = 6 + 3 = 9\\ &\text{Q5: } y = 3x + 4 \text{ for } x = 2 \implies y = 3(2) + 4 = 6 + 4 = 10\\ &\text{Q6: } 25 + 10h \text{ for } h = 4 \implies 25 + 10(4) = 25 + 40 = 65 \end{aligned}$$

\]

Simplify expressions for question 7

\[

$$\begin{aligned} &\text{a. } 3(x + 2) + 4(x - 2) = 3x + 6 + 4x - 8 = 7x - 2\\ &\text{b. } 2(x + y) - 3(x + y) = 2x + 2y - 3x - 3y = -x - y\\ &\text{c. } -(y + 8) + 4(6 - y) = -y - 8 + 24 - 4y = -5y + 16\\ &\text{d. } 2(4x + 3) - 3(x + 1) = 8x + 6 - 3x - 3 = 5x + 3 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(L + 3O\) (where \(L\) is the number of cups of lemonade and \(O\) is the number of cups of orange juice)

Question 2

\(14\)

Question 3

\(5\)

Question 4

\(9\) (Note: The question text contains a typo writing \(x = 12\) instead of \(z = 12\))

Question 5

\(10\)

Question 6

\(\$65\)

Question 7

a. \(7x - 2\)
b. \(-x - y\)
c. \(-5y + 16\)
d. \(5x + 3\)
</answer>

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Answer:

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<reasoning>

Formulate expression for drink sales

\[

$$\begin{aligned} &\text{Let } L = \text{number of cups of lemonade sold}\\ &\text{Let } O = \text{number of cups of orange juice sold}\\ &\text{Total earnings} = 1 \cdot L + 3 \cdot O = L + 3O \end{aligned}$$

\]

Evaluate expressions for questions 2 to 6

\[

$$\begin{aligned} &\text{Q2: } 2x - 4 \text{ for } x = 9 \implies 2(9) - 4 = 18 - 4 = 14\\ &\text{Q3: } -b + 8 \text{ for } b = 3 \implies -(3) + 8 = 5\\ &\text{Q4: } \frac{1}{2}z + 3 \text{ for } z = 12 \implies \frac{1}{2}(12) + 3 = 6 + 3 = 9\\ &\text{Q5: } y = 3x + 4 \text{ for } x = 2 \implies y = 3(2) + 4 = 6 + 4 = 10\\ &\text{Q6: } 25 + 10h \text{ for } h = 4 \implies 25 + 10(4) = 25 + 40 = 65 \end{aligned}$$

\]

Simplify expressions for question 7

\[

$$\begin{aligned} &\text{a. } 3(x + 2) + 4(x - 2) = 3x + 6 + 4x - 8 = 7x - 2\\ &\text{b. } 2(x + y) - 3(x + y) = 2x + 2y - 3x - 3y = -x - y\\ &\text{c. } -(y + 8) + 4(6 - y) = -y - 8 + 24 - 4y = -5y + 16\\ &\text{d. } 2(4x + 3) - 3(x + 1) = 8x + 6 - 3x - 3 = 5x + 3 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

\(L + 3O\) (where \(L\) is the number of cups of lemonade and \(O\) is the number of cups of orange juice)

Question 2

\(14\)

Question 3

\(5\)

Question 4

\(9\) (Note: The question text contains a typo writing \(x = 12\) instead of \(z = 12\))

Question 5

\(10\)

Question 6

\(\$65\)

Question 7

a. \(7x - 2\)
b. \(-x - y\)
c. \(-5y + 16\)
d. \(5x + 3\)
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