Algebra I Unit 3

Algebra I
Unit 3:

Creating Equations

Section 1: Solve One Variable Equations

How to solve linear equations

1.

Isolate the variable to one side of the equation

2.

Combine like terms

3.

Solve for the variable

Linear Equations Examples

1.

2.

How to Solve Quadratic Equations

●When equations are in the form (x-a)2=b where a and b are both
constants and b is a perfect square, the equation can be solved by taking
the square root of both sides then solving as it were a linear equation.

Quadratic Equations Examples

1.

2.

How to Solve an Exponential Equation

●To solve an exponential equation where both sides of the equation can be
written in terms of the same base, b, where b>0 and b ≠o, use the
following property… if bx=by then x=y.

Exponential Equations Examples

1.

2.

Section 2: Solve One Variable Inequalities

How to Solve One Variable Inequalities

●You solve them just like you would with one variable inequalities, other
than when you multiply or divide by a negative you flip the inequality sign.

Examples 1 and 2

1.

2.

Examples 3 and 4

3.

4.

Example 5

Example 6

Section 3: Create Quadratic Equations and

Inequalities

Example 1.

Bart is the owner of a baseball stadium. He has found that when he charges $25.00 per
baseball game ticket, the average game attendance is 8,500. In addition, Bart has found that
with every $2.00 increase in the baseball game ticket price, the average game attendance
decreases by an average of 50. Determine the equation that models Bart's revenue, R, in
terms of x, the number of $2.00 increases over $25.00.

Example 2.

Amberlee is adding a room to her home. She has a maximum space of 300 square feet. She
wants the length of the room to be 5 feet longer than the width. Determine the inequality that
models the area of the room, A, in terms of w, the width of the room, given the space
constraint.

Section 4: Create Linear and Exponential

Equations and Inequalities

Example 1.

Hunter's gym charges a base monthly fee of $28.25, plus $3.75 for each fitness class. Write an
equation that could be used to determine how much money Hunter paid his gym in one
month.

Example 2.

For her job, Gloria makes $12.00 an hour, but she spends $0.50 each day traveling to
and from work. Write an equation that could be used to determine Gloria's net income
for one day.

Example 3.

The band members at Smith High School must complete at least 1,500 total hours of community service
during the school year. This year, the band students decided to complete all of their community service
hours at a local animal shelter. The animal shelter offers 5-hour work shifts to the band members on
weekends. If during the first semester, the band had a total of 785 community service hours from working at
the shelter, write an expression to represent the number of shifts the band members need to complete
during the second semester of the school year.

Example 4.

Samantha is selling boxes of cookies for a school fundraiser, and she earns $5.00 for each box
of cookies that she sells. Suppose Samantha must earn at least $250.00 during the fundraiser.
She has already made $85.00 selling cookies. How many more boxes of cookies, b, must
Samantha sell to earn at least $250.00?

Example 5.

Each year, the local community center sponsors a ping pong tournament. Play starts
with 64 participants, and during each round, half of the players are eliminated. Write an
equation to find out after how many rounds, x, the winning player is declared.

Example 6.

Kellie has a large plastic container that she plans to use to collect pennies. On the first day,
she puts 5 pennies into the container. Then, the next day, she puts in double the number of
pennies that she put in the container the day before. Kellie continues this pattern for 10 days.
Write an equation to determine on what day Kellie puts $12.80 in pennies into the container.

Example 7.

Colin purchased a rare coin collection currently valued at $250. The value of the coin
collection increases each year by 20%. Write an inequality that could be used to determine the
number of years, t, after his purchase, that the value of the coin collection will be more than
four times its current value.

Example 8.

Kenneth purchased a plot of land valued at $25,000. The value of the land decreases
each year by 10%. Write an inequality to determine the number of years, t, after his
purchase, that the value of the land will be less than or equal to $10,000.

Section 5: Solve Equations and Inequalities

Examples 1 and 2

1.

Pete's Pizzeria charges a $2.50 delivery fee for all deliveries and $5.99 for each medium pizza
ordered, tax included. The following equation represents the total cost, y, for a customer who

orders x medium pizzas and has it delivered.What does the y-intercept represent?

2.
The linear model below shows the population of a town x years after 1980. According to the

model, what was the population in 1980?

Example 3

3. Maria is attending County Community College in the fall. Each semester, the community college
charges students $275.00 per class, plus $325.00 in fees. The function below represents Maria's
semester cost, C(x), given that she enrolls in x classes. Suppose Maria draws a graph to represent her
semester cost. Determine whether the graph is increasing or decreasing.

Example 4.

Juan received $200.00 from his grandmother for his birthday, so he decides to open a
non-interest-bearing savings account. Each month, he plans to deposit $8.00 into the account. The
function below represents the balance of Juan's account, A(x), after x months. Suppose Juan draws a
graph to represent the balance of his account. Describe the end behavior of the graph.

Example 5.

Anita won $500.00 at a cooking competition, so she decides to open an interest-bearing
savings account, which is compounded continuously at a fixed annual interest rate of 3.5%.
The function below represents the balance of Anita's account, A(x), after x months. What does
the y-intercept represent?

Example 6.

Jamie is purchasing a new car. She knows that the value of the car will decrease by 12% each
year from the date of purchase. The function below represents the value of the car, V(x), x
years after the date of purchase. Suppose Jamie draws a graph to represent the value of the
car. Determine whether the graph is increasing or decreasing, and describe the end behavior.

Example 7.

A college student launches a new social networking website. The number of users is increasing at a
rate of 27% per month from the website's launch. The function below represents the number of
users, N(x), x months after the website's launch. The college student wants to examine the number of
users on the website from 3 months after the website's launch to 6 months after the website's
launch. What is the minimum number of users on this interval?

Example 8.

A launcher throws a ball from the top of a building at an height of 48 feet. The ball reaches a
maximum height of 98 feet after a time of 10 seconds. The ball falls to the ground after 24
seconds. This can be modeled by the quadratic function given below. What is the rate of
change from the original position until the ball reaches its maximum height?

Example 9.

P(x) is a profit function where x represents the sales price of an item. Find the profit made
when the item is priced at $20.

Example 10.

Hunter's gym charges a base monthly fee of $28.23, plus $3.75 for each fitness class. If
Hunter attended seven fitness classes last month, how much did he pay the gym?

Example 11.

For her job, Gloria makes $12.00 an hour, but spends $0.50 each day traveling to and from
work. If her net income from yesterday was $71.50, how many hours did Gloria work that day?

Example 12.

Samantha is selling boxes of cookies for a school fundraiser, and she earns $5.00 for each box
of cookies that she sells. Suppose Samantha must earn at least $250.00 during the fundraiser.
She has already made $85.00 selling cookies. If this situation is modeled by the inequality
below, how many more boxes of cookies, b, must Samantha sell to earn at least $250.00?

Example 13.

Josephine is baking cupcakes for a bake sale. Each batch of cupcakes requires her to use 3
eggs, and there are 15 cupcakes in each batch. She has already baked one batch. If Josephine
has 21 eggs left, what is the maximum total number of cupcakes that she can bring to the
bake sale?

Example 14.

A population of trees is dying of a disease. Each year, four-fifths of the previous year's
population remains. If there are 23,416 trees this year, how many trees will remain after 8
years?

Example 15.

There are 100 grams of a radioactive substance. Half of the radioactive substance decays each
minute. Using the inequality below, determine the number of minutes, t, that the substance
would have to decay, t, before there are less than 6.25 grams of the substance remaining.

Example 16.

Henry tosses a ball into the air from the top of an 8 meter high hill. The path of the
ball can be given by the equation y = -x2 + 2x + 8, where y is the height of the ball in
meters, and x is the time since the ball was thrown.

Example 17.

Joanna is at a bakery. She is buying a cake for $17.50 and is trying to decide how many
cupcakes she wants to buy. Each cupcake costs $2.95. Which of the following graphs
represents the function that models Joanna's total cost, f(x), in terms on the number of
cupcakes she buys, x?

Example 18.

A rabbit-breeding farm had 20 rabbits. If the number of rabbits doubled every year,
which of the following graphs represents the number of rabbits in the farm by the end
of each year?

Example 19.

Mason is diving into a swimming pool from a height of 12 feet. He reaches a maximum height
of 15 feet at a distance of 1 foot from the diving board and lands at a distance of 3.24 feet
away from the diving board. Which of the following graphs represents the path of Mason's
dive?

Section 6: Create Systems of Equations and

Inequalities

Rules of Inequalities:

●Greater Than: Shade above the dashed line and the symbol opens
towards the larger number >

●Greater Than or Equal To: Shade above the solid line and the symbol
opens towards the larger number ≥

●Less Than: Shade below the dashed line and the symbol is closed off
towards the smaller number <

●Less Than or Equal To: Shade below the solid line and symbol is closed off
towards the smaller number ≤

Example 1.

At Happy Tails Animal Boarding, cat food and dog food is purchased weekly. Last week, Karen
purchased 6 bags of cat food and 9 bags of dog food for a total of $117. This week, she purchased 4
bags of cat food and 7 bags of dog food for a total of $87. Assuming the prices for cat and dog food
have not changed over the past two weeks, find the system of equations which can be used to
determine the cost of a bag of cat food, c, and the cost of a bag of dog food, d.

Example 2.

Sam spent $20 on candy, and Billy spent $15 on candy at the store. Sam also wants to buy
packages of cookies that are $4.50 each. Billy wants to purchase packages of a different type
of cookie that are $5.75 each. They both want to buy the same number of packages of
cookies, and want to spend the same total amount of money at the store. Create a system of
equations to model the situation above, and use it to determine if there is a viable solution.

Example 3.

A vendor at a winter carnival sold cups of hot chocolate and coffee. A cup of hot chocolate cost
$4.50, and a cup of coffee cost $2.25. The vendor could sell no more than 400 cups of hot chocolate
and coffee. At the end of the day, the amount in sales from the hot chocolate and coffee was less
than or equal to $1,125. Draw a graph to represent the number of cups of hot chocolate and the
number of cups of coffee sold for that particular day. Also, which of the following situations is a
non-solution to the system of linear inequalities described above?

A.

100 cups of hot chocolate and 150 cups of coffee

B.

175 cups of hot chocolate and 50 cups of coffee

C.

250 cups of hot chocolate and 100 cups of coffee

D.

50 cups of hot chocolate and 300 cups of coffee

Section 7: Rewrite Equations

Example 1.

Given the following equation, solve for r.

Example 2.

Given the following equation, solve for t.

Example 3.

Given the following equation, solve for c.

Example 4.

Given the following equation, solve for y.

Example 5.

Given the following equation, solve for h.