• In Activity 1, students must select an open or closed interval, locate the number, and choose the appropriate direction in order to demonstrate knowledge of the relationship between the inequality and the graphical representation.
• Activity 3 evaluates students’ understanding of the correspondence between the operation in the inequality and its inverse.
• Use informal assessment during class discussion and throughout the lesson to listen to student interactions, ask guiding questions, and suggest strategies and techniques.
• The Equations & Inequalities Worksheet revisions give students an opportunity to evaluate and revise their work.
• Students have an opportunity to evaluate their own understanding as well as their partner’s work in the think-pair-share activity.

Activity 1: Number-Line Warm-Up

For this warm-up activity, have students work in pairs. Have two sets of eight Inequality Cards and Number Line Cards (M-8-2-1_Inequality Cards.doc) for each pair. Prepare these by cutting apart Set A of the Inequality Cards and placing them in an envelope, cutting apart Set B and placing them in a separate envelope, and cutting apart the Number Line Cards so that each student has eight number line cards. One student in the pair will work with Set A, and the other will work with Set B.

The first step in this activity is for students to individually represent on a Number Line Card each of the equations or inequalities written on the Inequality Cards in their envelope.

Walk around the room as students are doing the activity. Once the number lines are drawn, have students place the Number Line Cards back in the envelope, along with the Inequality Cards. Students will now exchange envelopes with their partners. They will take the set of Inequality Cards and match them to the Number Line Cards that their partners just created. This allows students to check each other’s work, while at the same time giving them additional practice with number line representations of inequalities.

Choose several examples from the cards to review together as a group. Have one or two pairs come up to the board; ask one partner to write the inequality, and ask the other to draw the representation on a number line.

Ask students to think of a real-life situation involving an inequality and have one or more students write this inequality on the board and represent it on a number line. (Examples to prompt students could include ages less than the age required to get a driver’s license or the temperatures greater than the one at which water freezes, etc.) Now display on the board: 3x > 6 and x − 4 = 25.

Ask students: “What if we wanted to represent the solutions to these on a number line? How could we do that?” Provide time for some student responses/ideas, encouraging students to share their thinking processes.

“To be able to do this, we would first want to get our variable, x, alone on one side of the inequality or equal sign. In other words, we need to solve for x.”

Reaffirm any correct student responses and correct any misstatements or misconceptions. Be sure to emphasize correct vocabulary.

“Today we will learn how to solve one- and two-step linear equations and inequalities using our knowledge of inverse operations.”

Solving Equations—Review

Display and discuss the examples below or use similar examples. Emphasize how the inverse relationships of addition/subtraction and multiplication/division still hold true, even when variables are involved.

Examples showing how inverse operations can be used to isolate a variable:

• x + 3 = 6 and 6 − 3 = x
• 4y = 24 and 
• −7x = 14 and

“Looking at the examples we just did, what do you notice about the variable in each equation?” Encourage and discuss various student responses, focusing on the observation that using inverse operations in the proper order results in isolating the variable on one side of each equation.

Emphasize that isolating the variable is the goal in solving equations.

“Look at how to solve an equation for x using algebraic methods, keeping in mind what you have observed about inverse operations.”

Demonstrate the following steps on the board:

x − 4 = 25

x − 4 + 4 = 25 + 4       “Use an inverse operation to isolate the variable. Notice that the inverse of subtracting 4 is adding 4.

Write out the step to show that you do the same thing on both sides of the equal sign. This keeps the equation balanced.

x = 25 + 4                    Now simplify each side of the equation. Notice how the −4 and 4 cancel each other out because they are inverse numbers, leaving our variable alone.”

x = 29

Review the idea of the equal sign representing equality and balance. Remind them that to keep this balance students must make sure that whatever is done to one side of the equation is also done to the other side. Emphasize that this is true for any mathematical operation involving equations. Use the visual image of a balance scale and equal weights to help solidify the concept of balance.

Activity 2: Solving Equations Review

Distribute the Equations & Inequalities Worksheet (M-8-2-1_Inequalities Worksheet.docx and M-8-2-1_Inequalities Worksheet KEY.docx). Go over problems 1 and 2 together; then have students do problems 3–8 on their own.

After students complete these problems, discuss what finding a solution means:

“Remember, the solution to an equation is the value of the variable that makes the equation true. Luckily for us, this means we can always test our solutions by substituting the supposed value of the variable into the original equation and making sure the equation stays true.”

Instruct students to check each solution from problems 1–8 of the Equations & Inequalities Worksheet.

Solving Linear Inequalities

“As you know, equations are not the only types of mathematical sentences we can solve. In fact, we can use inverse operations to help us solve inequalities as well as equations! Let’s look again at the inequality 3x > 6.”

Display the following on the board. Go through each step with the class.

            “The inverse of multiplying by 3 is dividing by 3.

                         Simplify both sides of the inequality.

              We can now represent this solution on our number line.”

“As you can see, the algebraic techniques that we used to solve equations can also be used to solve inequalities. We can add, subtract, multiple, or divide both sides of an inequality by the same number without changing the value of the solution.”

Display on the board: 8 > 4

Have students experiment with doing several different operations to both sides of the inequality and seeing if the inequality still holds true. For example:

       “Dividing both sides by 2 yields 4 > 2, which is true.

        Adding 2 to both sides yields 10 > 6, which is true.

     Subtracting 9 from both sides yields −1 > −5, which

is true.”

“Did anyone experiment with an operation that made the resulting inequality NOT true?” (Here, you are looking to see whether or not a student attempted to multiply or divide both sides of the inequality by a negative number—the only way to obtain an untrue result. If a student has done this, allow this student’s example to lead into the next segment. If no student has done this, instruct all students to consider the following example.)

• 8 > 4               
• 8(−1) > 4(−1) 
• −8 > −4                 NOT TRUE!

“What do you think has happened here?” (Allow students to discuss. Lead them to an explanation similar to this example response: When we multiply or divide any number by a negative value, the sign of our original number changes. The nature of negative numbers is such that the larger the value of the negative, the smaller the true value of the number. For example, 5 > 2, but −5 < −2.)

“What do you think we need to do to properly multiply or divide both sides by a negative value?” (Switch the direction of the inequality symbol as soon as your solution process requires multiplying or dividing both sides by a negative value.)

Activity 3: Solving Inequalities

Instruct students to work together through problems 9 and 10 on the Equations & Inequalities Worksheet and have them complete problems 11–16 on their own.

Discuss solutions to inequalities as being the values for the variable that make the inequality true (just like with equations). Have students check this by taking their solutions from the Equations & Inequalities Worksheet and substituting different values for x into the original inequality to see if it holds true. For example, for problem 9, the solution is x > 2, so students will substitute several different values of x that are greater than 2 to test that the inequality stays true.

Two-Step Equations and Inequalities

“Notice how the solutions to the following examples change when these different symbols are used.”

Write these symbols on the board to help students look for them:

           “Subtract 2 from both sides of the equation.

  “Subtract 2 from both sides of the inequality.

          “Subtract 2 from both sides of the inequality.

           “Subtract 2 from both sides of the inequality.

           “Subtract 2 from both sides of the inequality.

“What is the difference in meaning between <, > and ≤, ≥?” (Less /greater than does not include the limit, while less/greater than or equal to does.)

“Because the meanings are different, when we are graphing the solutions to inequalities on a number line, we must be able to easily show when we are dealing with a <, > or a ≤, ≥ situation. We do this by using open and closed circles at the limit. An open circle is used if the limit is not included in the solution set, while a closed circle is used if the limit is included in the solution set.”

Project the following examples:

“Let’s look at some more examples of solving equations and inequalities.”

More examples:

          “Add 4 to each side of the equation.

“Use the same operations as tools in the same way to solve the inequality.”

          “Add 4 to each side of the inequality.

          “Add 4 to each side of the inequality.

          “Add 4 to each side of the inequality.

          “Add 4 to each side of the inequality.

               Divide both sides of the inequality by 12.”

“Now let’s graph the solutions to the equation and inequalities we just looked at. When graphing x = 1, we simply place a point above the 1 on the number line. When graphing the inequalities, we will follow the same guidelines as before.”

Give students time to graph the solutions to the previous examples. If additional practice is needed, you may also discuss solving and graphing the following examples:

Additional examples:

                  “Subtract 9 from each side of the equation.

                   “Subtract 9 from each side of the inequality.

                   “Subtract 9 from each side of the inequality.

                  “Subtract 9 from each side of the inequality.

                   “Subtract 9 from each side of the inequality.

At this point, initiate a think-pair-share activity to evaluate students’ level of understanding.

Display on the board and instruct students to solve the following for x. Also prepare students to compare/contrast the solution techniques used and graph each solution on a number line.

• (−3)x = 27                                           Answer:           x = −9
• (−3)x > 27                                           Answer:           x < −9

Have students take out a sheet of paper and complete the above problems on their own first, without discussing the solutions with others. (Allow several minutes for students to work on these problems.)

“Now, compare your solutions with your partner’s solutions and, if there are any differences, discuss them. By working together, come to a consensus. Check your answers by substituting a solution value into the original equation or inequality. Be ready to share your results with the rest of the class.”

Randomly call on two or three pairs to share their answers with the class. Emphasize checking the answers by substituting solution values into the original equation or inequality.

Extension:

• Students who demonstrate proficiency can work in pairs or small groups to generate real-world examples of simple equations and inequalities involving one variable. Provide chart paper and colored markers for students to present these scenarios to the rest of the class for additional practice.
• Small Groups: Use a think-pair-share activity to identify students who might need opportunities for additional learning. Provide additional practice if necessary, working in small groups to review problem solving step-by-step and to check solutions together.