Students will learn how to solve simple , one- or two-step linear equations and inequalities. They will: 
- reinforce their comprehension of the inverse operations of addition/subtraction and multiplication/division, and apply these rules to help them isolate a variable. 
- compare and contrast the solution techniques for equations vs. inequalities. 
- use the number line to highlight "greater than" and "less than" relationships. 
- determine which real-world situations could be described using these equations and inequalities. 
- solve one- and two-step equations and inequalities.

- How can mathematics help us communicate more effectively? 
- How are relationships represented mathematically? 
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems? 
- How may data be arranged and represented to reveal the relationship between quantities? 
 

- Coefficient: The coefficient of a term in an expression is the number that is multiplied by one or more variables or powers of variables in the term. 
- Function: A relation in which every input value has a unique output value. 
- Linear Equation: An equation of the form Ax + By = C, where A ≠ 0 and B ≠ 0. The graph of a linear equation is a straight line. 
- Rate of Change: The ratio of the change in the output value and change in the input value of a function. 
- Simultaneous Linear Equations: Two or more linear equations that share a set of variables and either exactly one solution, no solutions, or infinitely many solutions; often called a “system of equations.” 
- Slope: The measure of the steepness of a line; the ratio of the change in the y-values to the x-values of the linear function; rate of change. 
- y-intercept: The y-coordinate of the point where a line intersects the y-axis; the value of the function when x = 0.

- Inequality and Number Line Cards (M-8-2-1_Inequality Cards)
- Equations & Inequalities Worksheet (M-8-2-1_Inequalities Worksheet)
- envelopes to store cards
- chart paper

- In Activity 1, students must choose an open or closed interval, locate the number, and choose the appropriate direction to demonstrate their understanding of the inequality's relationship to the graphical representation. 
- Activity 3 assesses students' comprehension of the relationship between the operation in the inequality and its inverse. 
Informal assessment is used during class discussions and during the lesson to listen to student interactions, offer guiding questions, and recommend strategies and techniques. 
- The Equations & Inequalities Worksheet revisions allow students to examine and update their work. 
- In the think-pair-share activity, students can evaluate both their own comprehension as well as their partner's work.

Scaffolding, Active Engagement, Modeling, Explicit Instruction
W: Students learn to solve one- and two-step linear equations and inequalities sing knowledge of inverse operations (addition/subtraction and multiplication/division). 
H: The inverse relationships of addition/subtraction and multiplication/division still hold true, even when variables are involved. Number lines extended in both directions, right and left, highlight this point and offer visual reinforcement. 
E: Students may relate the equations and inequalities they manipulated and balanced to each graphical representation allows them to apply their own creation in a different form. Similarly, combining information from each graph strengthens both representations.
R: The graphing activities, equations, and inequalities sheet for this lesson encourages students to provide feedback and correct themselves. The paired and group activities foster discussion and learning from one another. When working on the practice sheet, students are encouraged to self-correct and revise. 
E: The step-by-step dissection of 5x + 2 = 12 gives students a tool with which they can create their own representations of adding or taking equal numbers from both sides of the equation. Each step is transparent, simple, quick, and allows for continual evaluation.
T: This lesson emphasizes the importance of understanding and applying inverse operations to solve equations and inequalities. When solving for a variable using algebraic techniques, make sure you do the same thing on both sides of an equation or inequality. Demonstrate the importance of applying inverse operations in preserving the truth of an equation or inequality.
O: This lesson aims to teach students how to solve simple one-step linear equations and inequalities using inverse operations. The activities are intended to help students progress from a basic grasp of inverse operations to applying that understanding to solve equations and inequalities. The number line is used to reinforce "greater than" and "less than" relationships, as well as to visually demonstrate the validity of solutions to an inequality for a variable. The practice of substituting a solution value into the original equation or inequality is highlighted to assist students in verifying their answers and gaining a better knowledge of the process.
 

Activity 1: Number-Line Warm-Up

For this warm-up activity, students should work in pairs. For each pair, use two sets of eight Inequality Cards and Number Line Cards (M-8-2-1_Inequality Cards). Prepare them by cutting out Set A of the Inequality Cards and putting them in an envelope, cutting out Set B and putting them in a separate envelope, and cutting out the Number Line Cards so that each student has eight number line cards. One student in the pair will work on Set A, while the other will concentrate on Set B. 

The first step in this activity is for students to independently represent each equation or inequality on a Number Line Card from the Inequality Cards in their envelope.

Walk around the room while the students are completing the activity. After drawing the number lines, instruct students to return the Number Line Cards and the Inequality Cards to the envelope. Students will now exchange envelopes with their partners. They will compare the set of Inequality Cards to the Number Line Cards that their partners have just generated. This allows students to review one other's work while also providing additional practice with number line representations of inequalities.

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

Ask students to consider a real-life situation with an inequality, and then have one or more students write the inequality on the board and illustrate it with a number line. (Examples to prompts students could include ages less than the age required to get a driver's license or the temperatures greater than the freezing point of water, etc.) Now show on the board that 3x > 6 and x - 4 = 25. 

Ask students, "What if we wanted to represent the answers to these on a number line? How could we do that?" Allow time for some student responses/ideas, allowing students to discuss their thinking processes.

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

Reaffirm any accurate student responses and correct any misstatements or misconceptions. Make sure to highlight the 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 following examples, or use similar ones. Emphasize how the inverse relationships of addition/subtraction and multiplication/division still hold true, even when variables are present. 

Examples of using inverse operations to isolate a variable:

x + 3 = 6 and 6 – 3 = x

4y = 24 and \(24 \over 4\) = y

-7x = 14 and \(14 \over (-7)\) = x

"Looking at the examples we just did, what do you notice about the variable in each equation?" Encourage and explore multiple student responses, emphasizing the fact that using inverse operations in the correct order results in isolating the variable on one side of each equation. 

Emphasize that the purpose of solving equations is to isolate the variable. 

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

Perform the following steps on the board:

x – 4 = 25

x – 4 + 4 = 25 + 4    "Use an inverse operation to isolate the variable. Note that the inverse of removing 4 is adding 4. 

Write out the steps to demonstrate that you do the same thing on both sides of the equal sign. This keeps the equation in balance.

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 concept of the equal sign, which represents equality and balance. Remind them that in order to maintain this balance, students must ensure that whatever is done to one side of the equation is also done to the other side. Make it clear that this applies to any mathematical operation involving equations. To help you understand the concept of balance, use the visual image of a balance scale with equal weights.


Activity 2: Solving Equations Review

Share the Equations & Inequalities Worksheet (M-8-2-1_Inequalities Worksheet). Go over problems 1 and 2 together, then have students complete problems 3-8 on their own.

After students finish these problems, talk about what finding a solution means:

"Remember that the solution to an equation is the value of the variable that makes the equation to hold true. Luckily for us, this means that we can always test our solutions by substituting the supposed value of the variable into the original equation and ensuring that the equation remains true." 

Instruct students to double-check their solutions to questions 1-8 on 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 solve both inequalities and equations! Let's review the inequality 3x > 6."

Show the following on the board. Discuss each step with the students.

3x > 6

\(3x \over 3\) > \(6 \over 3\)                 “The inverse of multiplying by 3 is dividing by 3.

x > \(6 \over 3\)                        Simplify both sides of the inequality.

x > 2                              We can now represent this solution on our number line.”

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

Display on the board: 8 > 4. 

Students should experiment with different operations on both sides of the inequality to see if the inequality still holds true. For example:

\(8 \over 2\) > \(4 \over 2\) →  4 > 2              “Dividing both sides by 2 yields 4 > 2, which is true.

8 + 2 > 4 + 2 → 10 > 6            Adding 2 to both sides yields 10 > 6, which is true.

8 – 9 > 4 – 9 → -1 > -5          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 determine if a student tried to multiply or divide both sides of an inequality by a negative number, which is the only way to get an incorrect result. If a student has done this, use their example to lead into the next part. If no student has done this, ask 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. Provide them with an explanation similar to this example response: When we multiply or divide a number by a negative value, the sign of the original number changes. The nature of negative numbers is such that the larger the negative value, the smaller the true value of the number. For instance, 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 whenever your solution procedure needs multiplying or dividing both sides by a negative number.) 


Activity 3: Solving Inequalities

Instruct students to work together on problems 9 and 10 of the Equations & Inequalities Worksheet, and then solve problems 11-16 on their own.
Discuss solutions to inequalities as being the values for the variables that make the inequality true (similar to equations). Students should test 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. For example, for problem 9, the solution is x > 2, thus students will replace numerous different values of x that are greater than 2 to check that the inequality holds true.


Two-Step Equations and Inequalities

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

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

=       <       ≤        >        ≥

5x + 2 = 12

5x = 10            “Subtract 2 from both sides of the equation.

x = 2                 Divide both sides of the equation by 5.”
__________________________________________________________________________

5x + 2 < 12

5x < 10            “Subtract 2 from both sides of the inequality.

x < 2                 Divide both sides of the inequality by 5.”
__________________________________________________________________________

5x + 2 ≤ 12

5x ≤ 10             “Subtract 2 from both sides of the inequality.

x ≤ 2                  Divide both sides of the inequality by 5.”
___________________________________________________________________________

5x + 2 > 12

5x > 10             “Subtract 2 from both sides of the inequality.

x > 2                  Divide both sides of the inequality by 5.”
___________________________________________________________________________

5x + 2 ≥ 12

5x ≥ 10              “Subtract 2 from both sides of the inequality.

x ≥ 2                   Divide both sides of the inequality by 5.”


"What is the difference in meaning between <, > and ≤, ≥?" (Less/greater than does not include the limit, whereas 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 drawing open and closed circles at the limit. An open circle is used if the limit is not included in the solution set; otherwise, use a closed circle if the limit is include in the solution set."

Consider the following examples:

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

More examples:

12x – 4 = 8
                                 “Add 4 to each side of the equation.
12x = 12            

x = 1                         Divide both sides of the equation by 12.”
_________________________________________________________________________________________

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

12x – 4 < 8
                                 “Add 4 to each side of the equation.
12x < 12 

x < 1                          Divide both sides of the inequality by 12.”
_________________________________________________________________________________________

12x – 4 ≤ 8
                                 “Add 4 to each side of the equation.
12x ≤ 12 

x ≤ 1                          Divide both sides of the inequality by 12.”
_________________________________________________________________________________________

12x – 4 > 8
                                 “Add 4 to each side of the equation.
12x > 12 

x > 1                          Divide both sides of the inequality by 12.”
_________________________________________________________________________________________

12x – 4 ≥ 8
                                 “Add 4 to each side of the equation.
12x ≥ 12 

x ≥ 1                          Divide both sides of the inequality by 12.”


"Now, let's graph the solutions to the equations and inequalities we just examined. When graphing x = 1, we simply place a point above the 1 on the number line. When graphing the inequalities, we will apply the same rules as before." 

Allow students time to graph the solutions to previous examples. If more practice is required, you might additionally discuss solving and graphing the following examples:


Additional examples:

-3x + 9 = -6
                             “Subtract 9 from each side of the equation.
-3x = -15 

x = 5                      Divide both sides of the equation by -3.”
_________________________________________________________________________________________

-3x + 9 < -6
                             “Subtract 9 from each side of the equation.
-3x < -15 

x > 5                      Divide both sides of the equation by -3 and switch the direction of the inequality symbol.”
_________________________________________________________________________________________

-3x + 9 ≤ -6
                             “Subtract 9 from each side of the equation.
-3x ≤ -15 

x ≥ 5                      Divide both sides of the equation by -3 and switch the direction of the inequality symbol.”
_________________________________________________________________________________________

-3x + 9 > -6
                             “Subtract 9 from each side of the equation.
-3x > -15 

x < 5                      Divide both sides of the equation by -3 and switch the direction of the inequality symbol.”
_________________________________________________________________________________________

-3x + 9 ≥ -6
                             “Subtract 9 from each side of the equation.
-3x ≥ -15 

x ≤ 5                      Divide both sides of the equation by -3 and switch the direction of the inequality symbol.”


At this point, start a think-pair-share activity to assess students' understanding. 

Display the following on the board and ask students to solve for x. Prepare students to compare or contrast solution strategies, as well as 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 solve the above puzzles on their own before discussing the solutions with others. Allow students a few minutes to work on these tasks. 

"Now compare your solutions to your partner's solutions and discuss any differences. Reach an agreement by working together. Check your answers by replacing the solution value into the original equation or inequality. Prepare to share your results with the rest of the class."

Randomly select two or three pairs to share their answers with the class. Check the answers by substituting the solution values into the original equation or inequality. 



Extension: 

Students who have demonstrated proficiency can work in pairs or small groups to create real-world examples of simple equations and inequalities with one variable. Provide chart paper and colored markers so that students can present these scenarios to the rest of the class for additional practice. 

Small Groups: Use a think-pair-share activity to identify students who may benefit from additional learning opportunities. If necessary, provide more practice by working in small groups to go over problem solving step by step and check solutions together.