Question

1. match each set of equations with the move that turned the first equation into the second.

a. $6x + 9 = 4x - 3$ $2x + 9 = -3$ 1. multiply both sides by $\frac{-1}{4}$
b. $-4(5x - 7) = -18$ $5x - 7 = 4.5$ 2. multiply both sides by -4
c. $8 - 10x = 7 + 5x$ $4 - 10x = 3 + 5x$ 3. multiply both sides by $\frac{1}{4}$
d. $\frac{-5x}{4} = 4$ $5x = -16$ 4. add $-4x$ to both sides
e. $12x + 4 = 20x + 24$ $3x + 1 = 5x + 6$ 5. add $-4$ to both sides

1. select all the situations for which only zero or positive solutions make sense.

a. measuring temperature in degrees celsius at an arctic outpost each day in january.
b. the height of a candle as it burns over an hour.
c. the elevation above sea level of a hiker descending into a canyon.
d. the number of students remaining in school after 6:00 p.m.
e. a bank account balance over a year.
f. the temperature in degrees fahrenheit of an oven used on a hot summer day.

Explanation:

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Problem 4

Step1: Analyze Set A

First equation: $6x + 9 = 4x - 3$
Second equation: $2x + 9 = -3$
Subtract $4x$ (add $-4x$) to both sides:
$6x + 9 - 4x = 4x - 3 - 4x$
$\implies 2x + 9 = -3$
Matches move 4.

Step2: Analyze Set B

First equation: $-4(5x - 7) = -18$
Second equation: $5x - 7 = 4.5$
Multiply both sides by $\frac{-1}{4}$:
$\frac{-1}{4} \times -4(5x - 7) = \frac{-1}{4} \times -18$
$\implies 5x - 7 = 4.5$
Matches move 1.

Step3: Analyze Set C

First equation: $8 - 10x = 7 + 5x$
Second equation: $4 - 10x = 3 + 5x$
Add $-4$ to both sides:
$8 - 10x - 4 = 7 + 5x - 4$
$\implies 4 - 10x = 3 + 5x$
Matches move 5.

Step4: Analyze Set D

First equation: $\frac{-5x}{4} = 4$
Second equation: $5x = -16$
Multiply both sides by $-4$:
$-4 \times \frac{-5x}{4} = -4 \times 4$
$\implies 5x = -16$
Matches move 2.

Step5: Analyze Set E

First equation: $12x + 4 = 20x + 24$
Second equation: $3x + 1 = 5x + 6$
Multiply both sides by $\frac{1}{4}$:
$\frac{1}{4}(12x + 4) = \frac{1}{4}(20x + 24)$
$\implies 3x + 1 = 5x + 6$
Matches move 3.

• A: Arctic temperatures in January can be negative, so negative solutions are valid.
• B: Candle height starts positive and decreases to 0; it cannot be negative, so only non-negative solutions make sense.
• C: A hiker can go below sea level (negative elevation), so negative solutions are valid.
• D: Number of students cannot be negative; it can only be 0 or positive, so non-negative solutions make sense.
• E: Bank accounts can have negative balances (overdrafts), so negative solutions are valid.
• F: Oven temperatures are always positive (Fahrenheit), so only non-negative solutions make sense.

Answer:

A. 4. Add -4x to both sides
B. 1. Multiply both sides by $\frac{-1}{4}$
C. 5. Add -4 to both sides
D. 2. Multiply both sides by -4
E. 3. Multiply both sides by $\frac{1}{4}$

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Problem 5