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Chapter 2: Linear Equations and Inequalities -1- 2.6: One-Variable Linear Inequalities Homework and Future Dates: Homework 1: Linear Inequalities Worksheet Homework 2: Barron’s Book p. 66 (1-6, 13-16) Show all work Homework 3: Barron’s Book p. 66 (7 - 12) Show all work Exam 1 Retest: Monday, 26 October either 7:30 AM or 2:50 PM in E112. This grade with be averaged with your 1st Exam 1 grade. Next Test (Exam 3): A Period: 1 November E Period: 2 November Inequality Symbols: less than greater than less than or equal to greater than or equal to not equal to The solution of an inequality includes any values that make the inequality true. Solutions to inequalities can be graphed on a number line using open and closed dots. Open dots vs. Closed dots When the inequality sign does not contain When the inequality sign contains includes an equality bar beneath it, the dot is open. an equality bar beneath it, the dot is closed, or shaded in. Graph of x 1 -3 -2 -1 0 1 Graph of x 1 2 3 Lesser -3 Greater -2 -1 0 1 2 0 1 2 3 Greater Graph of x 1 -3 3 Lesser -1 Lesser Graph of x 1 -3 -2 Greater -2 -1 0 Lesser 1 2 3 Greater Graph of x 1 -3 -2 -1 0 1 Lesser 2 3 Greater To Graph an Inequality, you must do the following: ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities 1. Create a number line 2. Place at least three numbers on the line (the solution number, zero and a third number) 3. If x is on the left of the inequality, move on to step 4. If x is on the right, flip the inequality. 4. Place an open or closed circle on the number 5. Shade to the side of the number that contains the solution set (draw your arrow in the direction of the inequality sign) ________________________________________________________________________ Example 1 Graph x 5 The statement x 5 says that x can take any value greater than -5, but x cannot equal -5 itself. We show this on a graph by placing an open circle at -5 and drawing an arrow to the right. The open circle at -5 shows that -5 is not part of the graph. Solution Example 2 Graph 3 x The statement 3 x means the same as ___________________. ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -2- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Solution Work on these examples with your groups: On the number line provided, graph each inequality: (a) x3 (b) x4 (c) 4 x ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -3- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities (d) 0x Solutions: (a) x3 (b) x4 (c) 4 x Flip: x 4 (d) 0 x Flip: x 0 ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -4- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Writing Equivalent Inequalities Just like when we solve equation, most of the time, we do the same operation to both sides of an inequality… An equivalent inequality results when The same quantity is added to, or subtracted from, both sides Both sides are multiplied or divided by the same positive quantity. But inequalities have a tick when doing multiplication and division with a negative number…. Both sides are multiple or divide by the same negative quantity and the direction of the inequality sign is reversed. For example 12 8 12 8 4 4 3 2 ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -5- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Solving a Linear Inequality Solve an inequality the same way that you solve an equality, except when you multiply or divide both sides by a negative number, you must turn the inequality sign around. 8 3x 17 8 8 3x 9 3 3 x 3 Example 1 Find the solution set of: 25 4x 13 Subtract 25: Divide each side by -4 and flip the inequality sign: Example 2 Find and graph the solution set of 1 2x x 6 ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -6- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Directions: Solve and graph the following inequalities on a separate sheet of paper. Homework: Barron’s Book p. 66 (1-6, 13-16) Show all work ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -7- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Union of Sets Example 1 Graph the solution set for x 1 2x 9 x 3x 1 7. Step 1: Solve each equations: 1 2x 9 3x 1 7 Step 2: Graph the solution set of x 1 2x 9, which is x 4 Step 3: Graph the solution set of x 3x 1 7, which is x 2 Step 4: Graph the intersection of the two solution sets. Step 5: Describe this interval with an inequality. ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -8- Chapter 2: Linear Equations and Inequalities 2.6: One-Variable Linear Inequalities Directions: Solve and graph the following inequalities on a separate sheet of paper. Homework: Barron’s Book p. 66 (7 - 12) Show all work ________________________________________________________________________ Extra help is available from 7:30-8:20 AM on T/Th mornings in E112. I can be found in E114 usually. Be sure to wait at least 5 min for me to come to E112. Late homework can be submitted during extra help with a parent/ guardian signature. -9-