ONE AND TWO STEP INEQUALITIES

INTRODUCTION

The objective for this lesson on One and Two Step Inequalities is, the student will solve and graph the solutions to one – and two – step inequalities in mathematical and real-world situations.

The skills students should have in order to help them in this lesson include, one- and two- step equations, integer operations, order of operations and the number line.

We will have three essential questions that will be guiding our lesson. Number one, how does the solution of the inequality x plus six is less than ten differ from the solution of the equation x plus six is equal ten? Number two, when do you use an open circle when graphing an inequality? A closed circle? Number three, how do you make an inequality a true statement when multiplying or dividing by a negative number? Explain your answer.

Begin by completing the warm-up for this lesson on solving equations in order to prepare for the lesson on One and Two Step Inequalities.

SOLVE PROBLEM – PART ONE INTRODUCTION

The SOLVE problem for this lesson is, Jennifer and three of her friends are going to a concert. The price of a ticket includes entrance to the concert, a CD, and T-shirt. The total cost of the tickets is more than forty-eight dollars. How could you represent the cost of one ticket?

In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, how could you represent the cost of one ticket?

Now that we have identified the question we need to put this question in our own words in the form of a statement. This problem is asking me to find the representation for the cost of one ticket.

During Part One of this lesson we will learn how to solve and graph one-step inequalities in order to complete this SOLVE problem at the end of Part One of the lesson.

INEQUALITY SYMBOLS

Identify the location of zero on the number line.

What type of integers are to the right of zero on the number line? They are the positive integers.

What type of integers are to the left of zero on the number line? Negative integers

Is six greater than or less than zero? Six is greater than zero because it is to the right of zero on the number line.

Is negative eight greater than or less than zero? Negative eight is less than zero because it is to the left of zero on the number line.

Identify the two values we are comparing. Two and five

What sign can we use between the two values to make a true statement? The less than symbol, <

How can we write this relationship using words? Two is less than five

ONE-STEP INEQUALITIES – ADDITION AND SUBTRACTION

Complete the equations for Problems one through four in the Equation column.

Equations One is x plus two is equal to four. We solve that by isolating the variable and subtracting two from both sides. Our solution is x equals two. We substitute in the value of x into our original equation and find that four is equal to four. Our equation is balanced and our solution is correct.

Number two, again you need to isolate the variable so we subtract two from both sides and our solution is x is equal to negative six. For the check we substitute in negative six into the original equation for the variable x and solve to find that our equation is balanced.

Equation three, x minus two is equal to four. We isolate the variable by using the opposite operation of addition and find our value of x is equal to six. We check that by substituting six for the variable in the original equation and balance our equation.

And number four, x minus two is equal to negative four. We solve by using the opposite operation of adding two to both sides. And our solution for our equation is x is equal to negative two. We check once again by substituting in the value of negative two for our variable and balancing our equation.

What is the meaning of an equation and its equal sign? The equal sign means that the values on both sides of the equation must be the same. The equation must be a true statement.

Look at the problem in the inequality column. How is this problem different from the problem in Column One? It has an inequality symbol instead of an equals sign.

What does the inequality symbol mean? The value on one side of the symbol is less than, less than or equal to, greater than, or greater than or equal to the value on the other side of the symbol.

Inequalities can be solved using the same process as the one used to solve equations and like an equation, an inequality must be a true statement.

What are the two things that we need to remember when solving an equation? Isolate the variable, balance the equation.

In solving inequalities we will also need to isolate the variable and whatever we do to one side of the inequality we must do to the other.

How did we find the solution for this equation? We had to isolate the variable by subtracting two from both sides.

What was the solution for this equation? x is equal to two

How can we apply what we know about solving equations to complete the first step in solving the inequality in Problem One? We can isolate the variable by subtracting two.

What will you need to do to the other side of the inequality? Also subtract two.

What is the value of x in the inequality? x must be less than two

How did we check the equation? We substituted the value for x back in the original equation.

We can check the answer for an inequality using the same process as the one we used to check the answer for an equation.

Take a look back at the equation from Problem One.

How many values were there for the variable x? One

In an inequality, there is more than one value that will make the statement true. Look at the inequality above.

What is the solution for the inequality in Problem One? x is less than two

What does this solution mean? Any value that is less than two should work when substituted back into the original inequality.

What values can we use to check the inequality? Any value that is less than two

Let’s use the value and substitute in the value of one. The value one, which is less than two, makes the statement true because three is less than four.

Choose a value that is greater than two, such as five, to try in the original inequality.

Five plus two, is that less than four? Seven is not less than four. So that is not a true statement because seven is not less than four.

ONE – STEP INEQUALITIES – MULTIPLICATION AND DIVISION

Complete the equations for Problems one through four in the Equation column.

Number One, two x is equal to four. This is a multiplication equation so we’ll solve using the opposite operation of division. We balance the equation by dividing both sides by two and our solution is x is equal to two. We substitute that value of x into our original equation to determine if it is balanced.

Number Two, two x is equal to negative four. Again, we use the opposite operation of division and balance our equation by dividing both sides by two. Our solution is x is equal to negative two. We check it by substituting in that value into our original equation to determine if it is balanced.

Number Three, x divided by two is equal to two. We use the opposite operation of multiplication to solve this equation. We multiply both sides by two to keep the equation balanced and our solution is x equals four. We substitute back into the original equation the value of four to determine if it is balanced.

Equation Four, x divided by two is equal to negative two. Again, we solve using the opposite operation of multiplication. We multiply both sides by two to keep the equation balanced and our solution is x is equal to negative four. We check it by substituting in that value into the original equation to determine if the equation is balanced.

What is the meaning of an equation and its equal sign? The equal sign mean that the values on both sides of the equation must be the same. The equation must be a true statement.

Look at the problem in the Inequality column. How is this problem different form the problem in Column One? It has an inequality symbol instead of an equals sign.

What does the inequality symbol mean? The value on one side of the symbol is less than, less than or equal to, greater than, or greater than or equal to the value on the other side of the symbol.

What are the two things that we need to remember when solving an equation? Isolate the variable and balance the equation.

In solving inequalities we will also need to isolate the variable and whatever we do to one side of the inequality we must do to the other.

How did we find the solution for this equation? We had to isolate the variable by dividing both sides by two.

How can we apply what we know about solving equations to complete the first step in solving the inequality in Problem One? We can isolate the variable by dividing by two.

What will we need to do to the other side of the inequality? Also divide by two.

What is the value of x in the inequality? x is less than two.

How did we check the equation? We substituted the value for x back in the original equation. We can check the answer for an inequality using the same process as the one we used to check the answer for an equation.

Look back at Problem One. How many values were there for the variable x? There was one.

In an inequality, there is more than one value that will make the statement true.

What is the solution for the inequality in Problem One? x is less than two.

What does this solution mean? Any value that is less than two should work when substituted back into the original inequality.

Under the “Check” for Problem One inequality, what values can we use to check the inequality? Any value that is less than two.

Let’s try one. Substitute the value of one back into the original inequality. Two times one is less than four. Two is less than four, which is a true statement.

Let’s go back and choose a value that is greater than two, such as five, to try in the original inequality. Two times five is ten, is that less than four? Ten is not less than four, so this is not a true statement because ten is not less than four.

GRAPHING INEQUALITIES

What is the solution for the inequality in Problem One? x is less than two.

We can use the solution of the inequality to determine how to number the number line and how to graph the solution.

The solution contains a positive two, so we can place a two in the middle of the number line and label the values to the left and right of the two.

Is the value of two a solution for the inequality? Explain your thinking. No, because x is less than two.

Begin graphing the inequality by drawing a circle above the two. The circle above the two is open because two is not included in the solution.

Which direction should the arrow point? Justify your answer. The arrow should start at the two and point to the left because all solutions are values less than positive two.

What does the arrowhead at the end of the ray indicate? The values for the solution will continue to infinity.

What is the solution for the inequality in Problem Two? x is greater than or equal to negative six.

We can use the solution of the inequality to determine how to number the number line and how to graph the solution.

The solution contains a negative six, so we can place a negative six in the middle of the number line and label the values to the left and right of the negative six.

Is the value of negative six a solution for the inequality? Explain your thinking. Yes, because x is greater than or equal to negative six.

The circle above the negative six is closed because negative six is included in the solution.

Which direction should the arrow point? Justify your answer. The arrow should start at the negative six and point to the right because all the solutions are values greater than negative six.

What does the arrowhead at the end of the ray indicate? The values for the solution will continue to infinity.

INEQUALITIES WITH NEGATIVE NUMBERS

Number One, four is less than seven. Is the number statement true or false? The statement is true.

When we were solving equations and inequalities and we used multiplication or division, what did we have to do? Whatever operation we used on one side of the equals sign or inequality sign, we had to do to the other.

What are we multiplying by in Problem One? A negative two

What is the product on the left side of the inequality? Negative eight

What is the product on the right side of the inequality? Negative fourteen

Is negative eight is less than negative fourteen true of false? It is false

Negative eight is not less than negative fourteen.

What do we know about two numbers written with an inequality sign? It must be a true statement.

What can we do to make it a true statement? We can flip the inequality sign.

How can we read the inequality after we flip the inequality sign? Negative eight is greater than negative fourteen or we can use the symbol negative eight is > negative fourteen.

Number Two, twelve is greater than three is our number statement and that is true.

What value are we dividing by in Problem Two? Negative three

What is the quotient on the left side of the inequality? Negative four

What is the quotient on the right side of the inequality? Negative one

Is negative four is greater than negative one true or false? It is a false statement.

Negative four is not greater than negative one.

What do we know about two numbers written with an inequality sign? It must be a true statement.

What can we do to make it a true statement? We can flip the inequality sign.

How can we read the inequality after we flip the inequality sign? Negative four is less than negative one or we can use the symbol negative four < negative one.

Complete Problems Three and Four.

When solving inequalities, you must switch the inequality symbol when you multiply or divide by a negative value.

INEQUALITIES – MULTIPLY AND DIVIDE WITH NEGATIVE NUMBERS

Let’s look at Problem One, negative two x is greater than four.

What do we need to do to isolate the variable? We need to divide both sides by negative two.

When we were dividing or multiplying by a negative value, what did we have to do to make the inequality true? We need to flip the inequality sign to make the inequality true. We divide both sides of our inequality by negative two and in order to make it a true statement x is less than negative two.

Check the answer by substituting in a solution into the original inequality. Try substituting negative four. What did you do? Since we have to divide by a negative to solve the inequality we flip the symbol to less than to make the inequality true.

Our solution for the inequality is x is less than negative two.

We draw our circle above the negative two, and it’s an open circle and our arrow points to the left to indicate that all of our solutions are less than negative two.

Problem Three, x divided by negative two is less than two. What do we need to do to isolate the variable? Multiply both sides by negative two.

When we multiply by a negative value, what do we need to do to make the inequality true? We need to flip it to the opposite symbol. We multiply both sides by negative two, and our solution is x is greater than negative four.

Check the answer by substituting in a solution into the original inequality. Let’s try substituting the value of two. What did you do? Since we had to multiply by a negative to solve the inequality we flipped the symbol to make the inequality true.

When we tried the value of two, negative one is less than two.

Now let’s graph the solution. x is greater than negative four. We draw an open circle above the negative four to indicate that, that value is not included in our solution set, and our arrow goes to the right to indicate that we want values that are greater than negative four.

SOLVE PROBLEM PART ONE – COMPLETION

Now let’s go back and look at our SOLVE problem from the beginning of the lesson. Jennifer and three of her friends are going too a concert. The price of a ticket includes entrance to the concert, a CD, and T-shirt. The total cost of the tickets is more than forty-eight dollars. How could you represent the cost of one ticket?

S, Study the Problem. We underlined the question and we completed this statement, this problem is asking me to find the representation for the cost of one ticket.

O, Organize the Facts. First we identify the facts. We go back and read our SOLVE problem again and mark each fact with a vertical line. Jennifer an three of her friends are going to a concert./ The price of a ticket includes entrance to the concert, a CD, and T-shirt./ The total cost of the tickets is more than forty eight dollars./ How could you represent the cost of one ticket?

After we identify the facts we go back and eliminate the unnecessary facts. In this problem, the price of a ticket includes entrance to the concert, CD, and T-shirt, is an unnecessary fact. Because it does not contain any information that will help us determine or represent the cost of one ticket.

After we eliminate the unnecessary facts then we list the necessary facts. Jennifer and her three friends for a total of four people, and the total cost is more than forty eight dollars.

L, Line Up a Plan. Write in words what your plan of action will be. Write and solve an inequality that I can use to represent the cost of one ticket.

Choose an operation or operations. Division

V, Verify Your Plan with Action. Estimate your answer. Our estimate here is that it will be more than ten dollars.

Then carry out your plan. We write an inequality four x is greater than forty-eight. We solve the inequality by dividing both sides by four. We find the value of x is greater than twelve.

E, Examine Your Results.

Does your answer make sense? Compare your answer to the question. Yes, because we were looking for how to represent the cost of one ticket.

Is your answer reasonable? Compare your answer to the estimate. Yes, because it is close to the estimate of more than ten dollars.

Is your answer accurate? Check your work. Yes.

Write your answer in a complete sentence. The cost of one ticket can be represented by the inequality x is greater than twelve dollars.

SOLVE PROBLEM PART TWO – INTRODUCTION

The SOLVE problem for Part Two of the lesson is, the art club is having a fundraiser to buy new art supplies. They want to raise at least one thousand five hundred sixty seven dollars. There are thirty-two students in the art club, and there is currently two hundred eight seven dollars in their art supply fund. The students have decided to divide the remaining financial need equally. How much will each student need to raise to meet the goal?

In Step S, we Study the Problem. First we need to identify where the question is located within the problem and underline the question. The question for this problem is, how much will each student need to raise to meet the goal?

Now that we have identified the question, we need to put this question in our own words in the form of a statement. This problem is asking me to find the amount each student will need to raise to meet the goal.

During this Part Two of the lesson we will learn how to solve and graph two-step inequalities to complete this SOLVE problem at the end of the lesson.

TWO-STEP INEQUALITIES

Identify the equation. It is a two-step equation.

Explain the meaning of a two-step equation. It is an equation that involves two operations to solve for the solution.

What were the steps when solving two-step equations? When solving two-step equations, we follow the order of operations (PEMDAS) in reverse order. Usually, multiplication and division are completed before addition and subtraction. However, in two-step equations, we will add or subtract before multiplying or dividing because we work toward isolating the operation with the variable.

Solve the two-step equation in Problem One. We add three to both sides to balance the equation. Then we divide both sides by two and our value of x is equal to five. We then complete the check by substituting in five for the value of the variable x. And our equation is balanced.

Two x minus three is less than seven.

How is this problem different from the equation? It has an inequality symbol instead of an equals sign.

How is solving an inequality similar to solving an equation? We can follow the same steps: we isolate the variable and whatever operation we apply to one side of the inequality, we must apply to the other.

What is the first step in solving the inequality? First we add three to both sides. Is my variable isolated? No. What operation do we do to isolate the variable? The coefficient is multiplied by the variable, so we use the inverse operation of division. x is less than five. Is my variable mow isolated? Yes.

Have I performed the same operations on both sides of the inequality? Yes.

What is the solution for the inequality? x is less than five

How did we check the solution for the one-step inequality? We chose a value that would be part of the solution set and substituted it back into the original inequality and solved to see if it was a true statement.

What is the suggested value to try in the Check? The value of positive one.

Is one less than the solution of x is less than five? Yes

What does this mean? It means that if we substitute the value of one in for x in the inequality, the answer should be a true inequality.

Substitute the value of one. Two times one minus three is less than seven. Two minus three is lesson than seven. Negative one is less than seven. That is true.

Now choose a value that is greater than five such as seven to try in the original inequality. Any value greater than five will make the inequality not true. This is not a true statement because eleven is not less than seven.

What is the solution for the inequality? x is less than five.

How do you model the solution to a one-step inequality? Use the value from the solution as the middle number on the number line scale. Draw a circle at the given value.

Is five a solution to the inequality? No, because x is less than five.

How do we model on the number line that five is not part of the solution set? We have an empty circle.

How do we know which direction the arrow should point for the graph? We need to extend the arrow from the five to the left because we want values than are less than five.

What does the arrowhead mean? The arrowhead at the end of the line shows that the values for the solution will continue to infinity.

TWO-STEP INEQUALITIES – MULTPLICATION AND DIVISION WITH NEGTIVE NUMBERS

Let’s review the relationship between positive and negative numbers and multiplying or dividing by a negative with inequalities.

Number One, negative seven is less than nine. That is a true statement.

When we multiply both sides of that inequality by the negative two we have fourteen is less than negative eighteen. That is a false statement. Fourteen is not less than negative eighteen.

What can I do to make it a true statement? I can flip the inequality sign. Fourteen is greater than negative eighteen or fourteen with the greater than sign (>) negative eighteen.

Fourteen is greater than two, is a true statement. We multiply or divide by the negative, this time we’re going to divide by negative two, and we have negative seven is greater than negative one.

Is that a true or a false statement? It is a false statement. Negative seven is not greater than negative one.

What can I do to make it a true statement? Flip the inequality sign. Negative seven is less than negative one.

Let’s look at an inequality with two-steps and negative numbers.

What is the first step in solving this inequality? We’re going to add four to both sides. We now have negative two x is less than or equal to twenty.

Have we isolated our variable? No

What is our next step? We need to divide both sides of the inequality by a negative two.

Let’s check our inequality. What is the suggested value to try? Four

Why would this be a good value to try to prove the inequality? Because it is greater than or equal to negative ten. We substitute the value of four in for x, complete the inequality and negative twelve is less than or equal to sixteen. So that is a true inequality.

Now let’s graph the solution. Because x is greater than or equal to negative ten, we’re going to start with our circle above the negative ten. The circle is filled in because we want to include the value of negative ten because x can be greater than or equal to negative ten. The arrow will point to the right, because all the values of x must be greater than or equal to negative ten. The arrowhead indicates that the answers will go through infinity.

WRITING AND SOLVING TWO-STEP INEQUALITIES IN REAL-WORD SITUATIONS

Jennifer is saving money to go to Adventure Camp during the summer. The cost of camp is three hundred five dollars, and she has saved fifty dollars. She is babysitting to earn money for camp. If she charges ten dollars per hour, what is the minimum number of hours she must babysit in order to have enough money for camp? Write an inequality to model the situation and then determine the solution.

What variable will we use to represent the hours in the word problem? h

What is on the left side of the inequality? The amount she started with

What other information will be written on the left side of the inequality? The variable times the charge per hour.

What will be written on the right side of the inequality? The total in her account

Let’s write the inequality. ten h plus fifty is greater than or equal to three hundred five.

What two operations are represented in Problem One? Multiplication and Addition

What operations will we use to solve the problem? Subtraction and Division

h is greater than or equal to twenty five and a half hours.

Let’s check using twenty-six hours.

Three hundred and ten is greater than or equal to three hundred and five. So it’s a true statement.

SOLVE PROBLEM PART TWO – COMPLETION

We are now going to go back to the SOLVE problem from the beginning of the lesson. The question was, the art club is having a fundraiser to buy new art supplies. They want to raise at least one thousand five hundred sixty seven dollars. There are thirty-two students in the art club, and there is currently two hundred eighty seven dollars in their art supply fund. The students have decided to divide the remaining financial need equally. How much will each student need to raise to meet the goal?

S, Study the Problem. Underline the question. This problem is asking me to find the amount each student will need to raise to meet the goal.

O, Organize the Facts. First we identify the facts. We go back to our original SOLVE problem and read it again. We draw a vertical line after each fact. The art club is having a fundraiser to buy new art supplies./ They want to raise at least one thousand five hundred sixty seven dollars./ There are thirty two students in the art club,/ and there is currently two hundred eighty seven dollars in their art supply fund./ The students have decided to divide the remaining financial need equally./ How much will each student need to raise to meet the goal?

After we identify the facts we eliminate the unnecessary facts. We go back to our SOLVE problem. We’re going to eliminate the first because it does not have any information we need to determine the amount each student will need to raise to meet this goal.

After we eliminate the unnecessary facts, we list the necessary facts. There is a need of at least one thousand five hundred sixty seven dollars. There are thirty-two students. They have two hundred eighty seven dollars. And they’re going to divide the need equally.

L, Line Up a Plan. Write in words what your plan of action will be. Write an inequality that I can use to solve the problem and then solve the inequality.

Choose an operation or operations. We are going to be using two operations, Subtraction and Division.

V, Verify Your Plan with Action. First estimate your answer. Our estimate here is about forty dollars.

Carry out your plan. Write the inequality, thirty two x plus two hundred eighty seven is greater than or equal to one thousand five hundred sixty seven. We isolate the variable first by subtracting the two hundred eighty seven from both sides, and then because this is a multiplication inequality we’ll solve by dividing. X is greater than or equal to forty dollars.

E, Examine Your Results.

Does your answer make sense? Compare your answer to the question. Yes, because we were looking for the amount each person would need to raise.

Is your answer reasonable? Compare your answer to the estimate. Yes, because it is close to the estimate of about forty dollars.

Is your answer accurate? Check your work. Yes.

Write your answer in a complete sentence. Each student will need to raise at least forty dollars.

CLOSURE

Now let’s go back and discuss the essential questions from this lesson.

Number one, how does the solution of the inequality x plus six is less than ten differ from the solution of the equation x plus six is equal to ten? An equation has only one solution, and an inequality has many solutions.

Number two, when do you use an open circle when graphing an inequality? When do you use a closed circle? Use an open circle when the value is not a solution to the inequality and a closed circle when the value is a solution to the inequality.

Number three, how do you make an inequality a true statement when multiplying or dividing by a negative number? Explain your answer. When multiplying or dividing by a negative value, flip the inequality sign to make the inequality true.