Mini-Lesson MA.912.A.3.5Solve linear equations and

inequalities• Symbolically represent and solve

multi-step and real-world applications that involve linear equations and inequalities.

Mini-Lesson MA.912.A.3.5

• Materials and Preparation:• Teacher – equipment for projecting the lesson,

examples, and practice problems. (This will vary depending on the equipment that you have available in your classroom.)

• Student – calculator, paper and pencil for taking notes and working problems

• Suggested link to videos: http://www.pinellas.k12.fl.us/lmt/resources.html

Note: just copy the link to your web browser – also, if you use Brain Pop the user name for login is your school name_hs and the password is bppinellas

Pre-Requisite Skills• Adding, subtracting, multiplying, and dividing

of fractions, decimals and whole numbers• Order of operations• Add and subtract integers• Add and subtract fractions• Evaluate expressions• Connect words and algebra• Compare and order real numbers• Distributive property• Solving one-step equations

Vocabulary

• Variable – A symbol used to represent a quantity that can change.

• Constant – Any value that does not change.• Equation – A mathematical sentence stating

that the two expressions have the same value.• Solution to an Equation – A value or values

that make the equation true.• Identity – An equation that is true for all values

of the variable.

Explicit Instruction:

Today we will explore the Essential Question, “What is the process for interpreting and solving real-world problems involving linear equations and linear inequalities?

In solving a multi-step equation or inequality:

• You solve a multi-step equation or inequality in the way same you solve a one-step, just adding a series of steps to solve.

• You solve a multi-step equation or inequality by reversing the order of operations.

• Make note - in solving an inequality if you multiply or divide by a negative number when isolating the variable remember to “flip” the sign.

Steps For Solving Real World Problems

Highlight the important information in the problem that will help you write your equation.

Define your variable(s).

Write your equation or inequality.

Use the steps for solving equations/inequalities.

Check your answer by substituting into the original equation/inequality.

Answer the question in the real world problem. Always write your answer in complete sentences!

Modeled Examples:

1. A salesperson's total salary includes a base pay of $500 per month plus 8.5% commission on the monthly sales. If x = the monthly sales and y = total salary, write a formula that can be used to determine his total salary for a month.

The commission is 8.5% of x or 0.085x.

The total salary, y, is the base pay plus commission.

The base pay is $500.

Therefore, y = 0.085x + 500.

2. A man plans to set up a stand at a flea market to sell hats. He will purchase several hats for a total of $75, and he will charge $6 for each hat he sells. In addition to the cost of the hats, he will need to pay $45 to set up the stand. Write an inequality that can be used to find out how many hats he must sell to make a profit of more than $60. His expenses are $75 for the hats plus $45 to set up the stand for a total of $120. His profit, P, is the income he makes after paying for his expenses.

His income in dollars from selling h hats is 6h.

Therefore, the inequality we can use is P > 6h – 120.

To solve: 60 > 6h – 120

+120 +120

180 > 6h

180/6 > 6h/6

30 > h

Therefore, the man must sell more than 30 hats to make a profit greater than $60.

1. Lori's weekly pay is $7.50 per hour for the first 40 hours and $10.75 per hour for each hour, h, over 40 that she works. Write an equation that Lori can use to find out how much money, M, she will make in dollars in a week if she works 42 hours.

Lori makes $7.50(40) or $300 for the first 40 hours of work. She makes $10.75h for each hour, h, over 40 that she works.

Therefore, M = 10.75(2) + 7.50(40)

And Lori’s money for the week is $321.50 for working 42 hours.

2. A car salesman is paid $6.50 per hour and $250 for each car that he sells. If he works h hours one week and sells c cars that week, what is a formula that can be used to determine his salary, s, in dollars for the week?

The salesman makes $6.50h for the number of hours, h, he works in a week.He makes $250c for the c cars that he sells in a week. Therefore, a formula for his salary, s, would be s = 6.5h + 250cSuppose this salesman wants to make at least $1,015 for this week with working only 30 hours. What is the minimum amount of cars he must sell in order to reach is goal? Use the equation above to now write your inequality and solve for c

s > 6.5h +250c

1,015 > 6.5(30) + 250c

1,015 > 195 + 250c

-195 -195

820 > 205c

820/205 > 205c/205

4 > c

Therefore, the salesman must sell at least 4 cars in order to make at least $1,015 for the week if he only 30 hours.

Guided Practice

Independent Practice Problem:

2. Karen works as a salesperson for a local marketing company. Using the equations shown below, the company calculates her monthly earnings based upon her total sales for the month. MONTHLY EARNINGS EQUATIONSTotal Sales for the Month Earnings Equation(s in dollars)

s < $5,000 E = 1,600 + 0.1s

s > $5,000 E = 1,600 + 0.1s + 0.15(s - 5000)

where:E represents total monthly earnings before taxes and withholdings represents the dollar amount of her total sales

Karen’s total sales were greater than $5,000 in October. If her total monthly earnings for October were $3,000, what was the value of her total monthly sales, s ?