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Solving Inequalities Student Probe At a Glance
Solve the inequality 2x  10  4 . Answer: x  3 Lesson Description This lesson is intended to help students develop an understanding of solving linear inequalities in one variable and interpreting the meaning of those solutions. It is strongly recommended that students be able to solve linear equations in a single variable with ease prior to this lesson. A preliminary lesson, Inequalities, is also recommended. Rationale Once students have achieved proficiency in solving linear equations in a single variable, they should be introduced to solving inequalities. Many real world problems involve expressions such as “at least”,” more than”, “at most”, and “less than”. Each of these requires an inequality to solve it. Preparation None Lesson The teacher says or does… 1. Solving inequalities is similar to solving equations. There are two differences that we need to think about. 2. First, let’s solve 2x  6 . What is the value of x? What: Solve inequalities Common Core State Standard: CC.9‐
12.A.REI.4 Solve equations and inequalities in one variable. Solve quadratic equations in one variable. Matched Arkansas Standard: AR.9‐
12.SEI.AI.2.1 (SEI.2.AI.1) Solve multi‐step equations and inequalities with rational coefficients: ‐‐ numerically (from a table or guess and check), ‐‐ algebraically (including the use of manipulatives), ‐‐ graphically, ‐‐ technologically Mathematical Practices: Look for and make use of structure. Who: Students who cannot solve inequalities Grade Level: Algebra 1 Prerequisite Vocabulary: inequality, solve, solution Prerequisite Skills: solve equations, order real numbers and expressions, use a number line Delivery Format: Individual, small group, whole group Lesson Length: 30 minutes Materials, Resources, Technology: None Student Worksheets: None Expect students to say or do… If students do not, then the teacher says or does… x 3
Two times what number is 6? The teacher says or does… Expect students to say or do… If students do not, then the teacher says or does… 3. Let’s model the solution on a number line. (Draw a number line with a closed circle at 3.) 4. How many numbers make Exactly 1 this statement true? 5. Now let’s solve 2x  6 . Think of this as an equation. 6. How many numbers make this statement true? Give me some examples. 7. Let’s model the solution on a number line. (Draw a number line with a closed circle at 3 and a ray extending to the left of 3.) 8. What do you see as one of the differences in solving equations and inequalities? (See Teacher Notes.) 9. Solve 2x  10  4 . x  3 An infinite number. Answers will vary. Does 0 make it a true statement? What about 5 ? An equation has one solution. Prompt students. An inequality can have an infinite number of solutions. 2 x  10  4
2 x  6 x 3
10. Give me some examples that make this statement true. 11. Let’s test x  6 . 2  6   10  4
Are there any other numbers that make the statement true? Think 2x  6 . Two times what number is 6? Answers will vary. (Example: x  6 .) Think 2x  10  4 . How would you solve it? Model for students. Refer to Solving Equations. Prompt students. Does x  6 make the statement true? 12. Let’s model the solution on a number line. Yes Students graph x  3 on the number line. The teacher says or does… Expect students to say or do… If students do not, then the 12  10  4 22  4
Is this statement true? Model. 13. We tested x  6 , but we can test any number greater than 3 and the statement will be true. 14. Solve 1  3x  7 . (Watch for the solution x  2 .) 15. Some of you got x  2 and some of you got x  2 . 16. Give me some examples and we will test to see which one is correct. (Examples: x  3, x  0 ) 17. For x  3 , 1  3  3   7
Answers will vary. Prompt students. 19 7
teacher says or does… 1  3x  7
 3x  6 x  2
Think 1  3x  7 . If students get the solution x  2 , refer to Inequalities. 10  7
That’s true! 18. For x  0, 1  3 0   7
10 7 17
That is not true! 19. Remember the property that we discovered. Multiplication or division by a negative number reverses the inequality. This is the second thing that makes solving inequalities different from solving equations. 20. Let’s model the solution Students graph x  2 on the on a number line. number line. x
x
21. Solve  2 . 2
4
4
x  8
Refer to Inequalities. Model. The teacher says or does… Monitor students. If students continue to fail to reverse the inequality, refer to Inequalities. Expect students to say or do… If students do not, then the 22. Give me some examples that make this statement true. (Test students’ examples.) Now model the solution on a number line. 23. Repeat with additional inequalities if necessary. Example: x  4 4
2
4
1  8
Students graph x  8 on the number line. teacher says or does… Model. Teacher Notes: 1. While there are exceptions, linear equations in a single variable usually have one solution. The idea here is for students to realize that inequalities can have an infinite solution set. 2. It is strongly recommended that Inequalities be taught as a preliminary lesson to this lesson. Inequalities addresses the property If a  b , then a  b . 3. Emphasis should be placed on graphing the solution set of the inequality. 4. Prior to this lesson students should be able to solve linear equations in a single variable with ease. Variations None Formative Assessment Solve the inequality: 3x  4  8 . Answer: x   4 References Paulsen, K., & the IRIS Center. (n.d.). Algebra (part 2): Applying learning strategies to intermediate algebra. Retrieved on September 11, 2011.