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?