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Introduction to One-Variable Inequalities – Understanding inequalities – Inequalities show the relative value of two expressions. – Recap: one-variable equations – Review solving for x in equations like x + 2 = 6. – Equations vs. inequalities – Unlike equations, inequalities express a range, not just one solution. – Why inequalities are important | Begin the lesson by explaining the concept of inequalities, emphasizing that they show how one value is less than, greater than, or not equal to another. Recap the previous lesson on solving one-variable equations to ensure a solid foundation for understanding inequalities. Highlight the differences between equations and inequalities, noting that while equations have one solution, inequalities represent a range of possible solutions. Stress the importance of inequalities in real-world situations, such as determining budget limits or minimum requirements. This will set the stage for learning how to solve two-step inequalities.

Understanding Two-Step Inequalities – Define a two-step inequality – An inequality solved with two operations, e.g., 3x + 5 > 11 – Compare to one-step inequalities – One-step: single operation; Two-step: involves two operations – Real-life examples – Budgeting: Income – (Expenses + Savings) > 0 – Solving two-step inequalities – Isolate variable: subtract/add, then divide/multiply | A two-step inequality requires two operations to solve, typically involving addition/subtraction and multiplication/division. Unlike one-step inequalities that involve only one operation, two-step inequalities provide a more complex challenge and are more common in real-life situations, such as budgeting or measuring. When solving, the goal is to isolate the variable on one side of the inequality. It’s crucial to perform the same operation on both sides of the inequality to maintain balance. After isolating the variable, discuss how the inequality symbol may change when multiplying or dividing by a negative number. Provide practice problems that include both positive and negative coefficients and constants.

Solving Two-Step Inequalities – Part 1 – Isolate the variable in the inequality – Identify the variable and get it alone on one side – Use inverse operations for constants – Subtract or add to remove the constant term – Example: Solve ‘3x + 4 > 10’ – Subtract 4 from both sides: 3x > 6, then divide by 3: x > 2 | This slide introduces the first steps in solving two-step inequalities, which are foundational for understanding how to find the range of solutions. Start by isolating the variable on one side of the inequality. This often involves performing inverse operations to eliminate the constant term. For example, in the inequality ‘3x + 4 > 10’, we first subtract 4 from both sides to get ‘3x > 6’. Next, we divide both sides by 3 to isolate x, resulting in ‘x > 2’. It’s crucial to remind students that the inequality sign stays the same when adding or subtracting but will flip when multiplying or dividing by a negative number. Encourage students to practice with various examples and check their solutions by substituting values for the variable.

Solving Two-Step Inequalities – Part 2 – Perform inverse operations – To isolate x, do the opposite of addition or multiplication – Check solution by substitution – Substitute the found value of x to see if the inequality holds true – Example: Solve 3x + 4 > 10 – Subtract 4 from both sides, then divide by 3 to find x – Verify solution with original inequality – Plug x value into ‘3x + 4 > 10’ to confirm | This slide continues the process of solving two-step inequalities. Emphasize the importance of performing inverse operations to isolate the variable. For example, to solve ‘3x + 4 > 10’, students should subtract 4 from both sides and then divide by 3. It’s crucial to check the solution by substituting the value of x back into the original inequality to ensure it makes the statement true. This step verifies their solution. Provide additional practice problems for students to solidify their understanding of these concepts.

Graphing Two-Step Inequalities – Graph solutions on a number line – Plot the range of solutions for the inequality – Open vs. closed circles – Open circle: > or 5 – Practice with different inequalities | This slide introduces students to the process of graphing two-step inequalities on a number line. Begin by explaining that the solution to an inequality is a range of values, which can be represented visually on a number line. Discuss the difference between open and closed circles, emphasizing that open circles represent ‘greater than’ or ‘less than’ (non-inclusive), while closed circles represent ‘greater than or equal to’ or ‘less than or equal to’ (inclusive). Demonstrate this concept with an example, such as x + 3 > 5, showing students how to isolate the variable and then graph the solution. Finally, encourage students to practice with different inequalities to solidify their understanding. Provide additional examples and practice problems for students to work on as homework.

Practice: Solving Two-Step Inequalities – Work on practice problems as a class – Attempt problems independently – Encourage self-reliance and problem-solving skills – Review solutions collectively – Discuss the correct steps and where mistakes occurred – Analyze common solving errors – Highlight frequent misconceptions and how to avoid them | This slide is focused on engaging students in active practice of solving two-step inequalities. Start by solving a few problems together as a class to demonstrate the process. Then, allow students to work on problems independently to build confidence. Afterward, come together to discuss the solutions, emphasizing the importance of each step in the process. Address common mistakes such as incorrect sign reversal or distribution errors, and provide strategies to avoid these pitfalls. This collaborative and reflective approach helps reinforce learning and correct misunderstandings.

Class Activity – Solve & Graph Two-Step Inequalities – Pair up and solve inequalities – Graph solutions on a number line – Use a number line to visualize the range of possible solutions – Present a solution to the class – Share your approach and answer with peers – Discuss solution methods – Compare different solving techniques | This interactive class activity is designed to reinforce the concept of solving two-step inequalities. Students will work in pairs to encourage collaboration and peer learning. Each pair will tackle a set of inequalities, providing an opportunity to practice and solidify their understanding. Graphing on a number line will help students visualize the solution set. Presenting to the class will build communication skills and confidence in their mathematical reasoning. As a teacher, facilitate the activity by providing guidance and ensuring each pair understands the steps involved. Possible activities for different pairs could include solving inequalities with different operations, using variables on either side of the inequality, or incorporating real-world contexts into the problems.

Wrapping Up: Two-Step Inequalities – Review of solving inequalities Remember to inverse operations and balance the inequality – Homework: Practice problems Solve 10 two-step inequality problems from the worksheet – Next topic: Multi-step inequalities We’ll explore more complex inequalities next class – Keep practicing and asking questions | As we conclude today’s lesson on two-step inequalities, remind students of the importance of performing inverse operations to both sides of the inequality to maintain balance. For homework, assign a set of problems that reinforce today’s learning, ensuring students are comfortable with the concept. Let them know that the next class will build on this foundation as we move into multi-step inequalities. Encourage them to practice the assigned problems and to reach out with any questions they might have before the next class.