Linear Expressions, Equations, &

Question

Linear Expressions, Equations, &
Inequalities Portfolio Worksheet
Directions: Use this worksheet to record your answers to the questions or
problems for the Linear Expressions, Equations, & Inequalities Portfolio. When you
are finished, save this worksheet with your answers and submit it for a portfolio
grade.
1. Roger is a wildlife researcher who has been hired to study and track the
population of bears in a national park. In Year 1 of the study, Roger determines
that the park is home to x bears. In Year 2, the population has increased to
x+10 bears, and in Year 3, the population has increased to 2(x+10) bears.
a. Use the Distributive Property to rewrite the bear population in Year 3
without needing to use parentheses.
b. The total number of bears in the park during Year 2 is 42. Write and solve a
linear equation to determine how many bears were in the park during Year
1. Explain the answer.
c. The total number of bears in the park during Year 3 is 50. Write and solve a
linear equation to determine how many bears were in the park during Year
1. Note that this answer will not be the same as the answer you got in part
b.
d. During Year 4, bad weather keeps Roger from finding the exact number of
bears in the park. However, he knows that y+5>38, where y is the number
of bears in the park during Year 4. Solve the inequality, and give three
possible values for y.
2. Jin is a wildlife researcher hired to track the population of wolves in the same
national park. In Year 1 of the study, there are z wolves; in Year 2 of the
study, there are 7+z wolves; and in Year 3 of the study, there are 3z-3 wolves.
The number of wolves in Year 2 is equal to the number of wolves in Year 3.
a. Write and solve a linear equation to determine the value of z.
b. Jin writes an expression to describe the number of wolves in Year 4 of the
study. The expression is (x+a)(x-b), where a and b are whole numbers
greater than 1 and less than 10. Write an expression that could be the
expression Jin uses to describe the number of wolves in the park during
Year 4 of the study. Then multiply the two terms of the expression.

GPT-4o mini · Answer

## Linear Expressions, Equations, & Inequalities Portfolio Worksheet ### 1. Bear Population Study a. Rewrite the bear population in Year 3 using the Distributive Property: $$ 2(x + 10) = 2x + 20 $$ So, the bear population in Year 3 is $2x + 20$. b. The total number of bears in the park during Year 2 is 42. To find the number of bears in Year 1: Given Year 2: $x + 10 = 42$ Solve for $x$: $$ x + 10 = 42 $$ $$ x = 42 - 10 $$ $$ x = 32 $$ **Answer:** There were 32 bears in the park during Year 1. This means that the population increased by 10 in Year 2. c. The total number of bears in the park during Year 3 is 50. To find the number of bears in Year 1: Given Year 3: $2(x + 10) = 50$ Solve for $x$: $$ 2(x + 10) = 50 $$ $$ x + 10 = 25 $$ $$ x = 25 - 10 $$ $$ x = 15 $$ **Answer:** There were 15 bears in the park during Year 1. This is different from part b because the growth in population may vary based on different yearly recordings or growth rates. d. Solve the inequality $y + 5 > 38$: $$ y + 5 > 38 $$ $$ y > 38 - 5 $$ $$ y > 33 $$ **Three possible values for $y$:** 34, 35, and 36. ### 2. Wolf Population Study a. Write and solve a linear equation for the number of wolves: Set Year 2 equal to Year 3: $$ 7 + z = 3z - 3 $$ Solve for $z$: $$ 7 + z + 3 = 3z $$ $$ 10 = 3z - z $$ $$ 10 = 2z $$ $$ z = 5 $$ **Answer:** The value of $z$ is 5, meaning there were 5 wolves in Year 1. b. To write an expression for the number of wolves in Year 4, we can choose $a = 3$ and $b = 4$, leading to the expression: $$ (x + 3)(x - 4) $$ Multiplying the two terms: $$ x^2 - 4x + 3x - 12 = x^2 - x - 12 $$ **Final expression for Year 4:** $x^2 - x - 12$ (the number of wolves in Year 4). --- Make sure to fill in any additional details or specific formatting as required by your instructor before submitting the worksheet.