Warm Up Andrew is practicing for a tennis tournament and needs more tennis balls. He bought 10 cans of tennis balls online and received a 25% discount. The shipping cost was $5.99. Let x represent the cost of each can. Write an algebraic expression to represent the cost of the tennis balls. Write an algebraic expression to represent the cost of the tennis balls with the discount. Write an algebraic expression to represent the total cost of the tennis balls with the shipping cost and the discount. Simplify the expression.

Creating Equations and Inequalities

Vocabulary Equation: a mathematical sentence that uses an equal sign (=) to show that two quantities are equal Linear equation: an equation that can be written in the form ax + b = c Rate: a ratio that compares different kinds of units

Creating Equations from Context Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the equation. Convert the appropriate units if necessary.

Example 1: James earns $15 per hour as a teller at a bank Example 1: James earns $15 per hour as a teller at a bank. In one week he pays 17% of his earnings in state and federal taxes. His take-home pay for the week is $460.65. How many hours did James work? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the equation. Convert the appropriate units if necessary.

Example 2: Brianna has saved $600 to buy a new TV Example 2: Brianna has saved $600 to buy a new TV. If the TV she wants costs $1,800 and she saves $20 a week, how many years will it take her to buy the TV? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the equation. Convert the appropriate units if necessary.

Example 3: Suppose two brothers who live 55 miles apart decide to have lunch together. To prevent either brother from driving the entire distance, they agree to leave their homes at the same time, drive toward each other, and meet somewhere along the route. The older brother drives cautiously at an average speed of 60 mph. The younger brother drives faster, at an average speed of 70 mph. How long will it take the brothers to meet each other? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the equation. Convert the appropriate units if necessary.

Example 4: Think about the following scenarios Example 4: Think about the following scenarios. In what units should they be reported? Explain the reasoning. Water filling up a swimming pool. The cost of tiling a kitchen floor. The effect of gravity on a falling object. A snail traveling across the sidewalk. Painting a room.

Example 5: Ernesto built a wooden car for a soap box derby Example 5: Ernesto built a wooden car for a soap box derby. He is painting the top of the car blue and the sides black. He already has enough black paint, but need to buy blue paint. He needs to know the approximate area of the top of the car to determine the size of the container of blue paint he should buy. He measured the length to be 9 feet 11 ¼ inches, and the width to be ½ inch less than 3 feet. What is the surface area of the top of the car? What is the most accurate area Ernesto can use to buy his paint? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the equation. Convert the appropriate units if necessary.

You Try: To celebrate graduation, you and 4 of your closest friends have decided to take a 5-day white-water rafting and hiking trip. During your 5-day trip, 2 days are spent rafting. If the rafting trip covers a distance of 60 miles and you are expected to raft 8 hours each day, how many miles must you raft each hour? For the hiking portion of your trip, you and your friends carry the same amount of equipment which works out to be 35 pounds of equipment each. For extra money, you can hire an assistant, who will carry 50 pounds of equipment. Each assistant charges a flat fee of $50 and an additional $22 for each mile. The total amount you would have to pay the assistant is $512. How many miles will your group be hiking? Is it worth hiring two assistants to help you and your friends carry the equipment? Justify your answers.

Warm Up Two people can balance on a seesaw even if they are different weights. The balance will occur when the following equation is satisfied. In this equation, w1, is the weight of the first person, d1 is the distance the first person is from the center of the seesaw, w2, is the weight of the second person, d2 is the distance the second person is from the center of the seesaw. Eric and his little sister Amber enjoy playing on the seesaw at the playground. Amber weighs 65 pounds. Eric and Amber balance perfectly when Amber sits about 4 feet from the center and Eric sits about 2 ½ feet from the center. About how much does Eric weigh? Their little cousin Aleah joins them and sits right next to Amber. Can Eric balance the seesaw with both Amber and Aleah on one side, if Aleah weighs about the same as Amber? If so, where should he sit? If not, why not?

Inequalities

Vocabulary Inequalities are similar to equations in that they are mathematical sentences, but they have infinite solutions, instead of only having one solution. Symbols used:

Solving Inequalities Solving a linear inequality is similar to solving a linear equation. The processes used to solve inequalities are the same as solving linear equations Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 1: Juan has no more than $50 to spend at the mall Example 1: Juan has no more than $50 to spend at the mall. He wants to buy a pair of jeans and some juice. If the sales tax on the jeans is 4% and the juice with tax costs $2, what is the maximum price of jeans Juan can afford? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 2: Alexis is saving to buy a laptop that costs $1,100 Example 2: Alexis is saving to buy a laptop that costs $1,100. So far she has saved $400. She makes $12 an hour babysitting. What’s the least number of hours she needs to work in order to reach her goal? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 3: A radio station is giving away concert tickets Example 3: A radio station is giving away concert tickets. There are 40 tickets to start. They give away 1 pair of tickets every hour for a number of hours until they have at most 4 tickets left for a grand prize. If the contest runs from 11:00 AM to 1:00 PM each day, for how many days will the contest last? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

You Try: The time has come for you to open a checking account You Try: The time has come for you to open a checking account. A local bank is offering you a free checking account if you maintain a minimum balance of $200. You already have a savings account with this bank and you have $60 saved. You decide to keep saving money until you have enough to open a checking account, plus keep some money in savings. If you deposit $15 a week into the savings account, what is the minimum number of weeks it will take for you to be able to open a checking account with at least $200 and still have $25 in your savings account?

Warm Up This year, Zachary has been babysitting his young cousins after school for $70 a month. His uncle also gave him an extra bonus of $100 for his excellent work. Since school started, Zachary has earned more than $500. How many months ago did school start? Write an inequality that represents the situation. Solve it and show all your work.

Vocabulary and Key Concepts Exponential Equations: equations that have the variable in the exponent The general form of an exponential equation is The base, b, must always be greater than 0 If the base is greater than 1, then the exponential equation represents exponential growth If the base is between 0 and 1, then the exponential equation represents exponential decay Another form of an exponential equation is

The pieces of the equation a: the initial value – look for words such as initial or starting r: rate of growth or decay Use (1+r) for exponential growth problems Use (1 – r) for exponential decay problems Look for words such as double, triple, half, quarter – which give the value for the base

Example 1: A population of mice quadruples every 6 months Example 1: A population of mice quadruples every 6 months. If a mouse nest started out with 2 mice, how many mice would there be after 2 years? Write an equation and then use it to solve the problem. Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 2: In sporting tournaments, teams are eliminated after they lose. The number of teams in the tournament decreases by half with each round. If there are 16 teams left after 3 rounds, how many teams started out in the tournament? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 3: The population of a small town is increasing at a rate of 4% per year. If there are currently about 6,000 residents, about how many residents will there be in 5 years at this growth rate? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

Example 4: You want to reduce the size of a picture to place in a small frame. You aren’t sure what size to choose on the photocopier, so you decide to reduce the picture by 15% each time you scan it until you get it to the size you want. If the picture was 10 inches long at the start, how long is it after 3 scans? Read the problem. Reread the scenario and make a list of the known quantities. Read the statement again and look for the unknown or the variable. Create expressions and inequalities from the known quantities and variable(s). Solve the problem. Interpret the solution of the inequality in terms of the context of the problem.

You Try: On opposite sides of a major city two suburban towns are experiencing population changes. One town, Town A, is growing rapidly at 5% per year and has a current population of 39,000. Town B has a declining population at a rate of 2% per year. Its current population is 55,000. Economists predict that in 5 years the populations of these two towns will be about the same, but the residents of both towns are in disbelief. The economists also claims that ten years after that, Town A will double the size of Town B. Can you verify the predictions based on the data given? Do you think these predictions will come true?