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000200010271713722_CH03.qxd 1/30/13 10:07 AM Page 92 Name Class Date Reteaching 3-1 Graphing and Writing Inequalities Two expressions separated by an inequality sign form an inequality. An inequality shows that the two expressions are not equal. Unlike the equations you have worked with, an inequality has many solutions. The solutions of an inequality are the values that make the inequality true. They can be graphed on a number line. Use a closed circle (•) for ⱕ and ⱖ and an open circle (o) for ⬎ and ⬍. For example: -5 -4 -3 -2 -1 0 1 2 3 x ⬍ -2 (“less than”) 4 -5 -4 -3 -2 -1 0 5 x ⱖ -2 (“greater than or equal to”) -5 -4 -3 -2 -1 0 1 2 3 4 1 2 3 4 5 x ⱕ -2 (“less than or equal to”) -5 -4 -3 -2 -1 0 5 1 2 3 4 5 Graph the inequality x S 4. All rights reserved. x ⬎ -2 (“greater than”) The inequality is read as “x is greater than 4.” Since all numbers to the right of 4 are greater than 4, you can draw an arrow from 4 to the right. Since 4 is not greater than itself, use an open circle on 4. -2 -1 0 1 2 3 4 5 6 7 8 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1. Graph the inequality x K ⴚ3. a. Write the inequality in words. b. Will the circle at ⫺3 be open or closed? c. Graph the solution. -7 -6 -5 -4 -3 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 5 6 2. Graph the inequality x R 3. a. Write the inequality in words. b. Will the circle at 3 be open or closed? c. Graph the solution. Course 2 Lesson 3-1 -4 -3 Reteaching 000200010271713722_CH03_L02.qxd 1/30/13 8:56 PM Name Page 2 Class Reteaching 3-2 Date Solving Inequalities by Adding or Subtracting To solve an inequality you can add the same number to or subtract it from each side of the inequality. Solve y ⫺ 3 ⬍ 2. Graph the solution. x⫹5ⱖ9 x ⫹ 5 ⫺ 5 ⱖ 9 ⫺ 5 Subtract 5 from each side. xⱖ4 Simplify. y⫺3⬍2 y⫺3⫹3⬍2⫹3 y⬍5 Add 3 to each side. Simplify. Graph: Graph: -6 -4 -2 Graph the solution. 0 2 4 6 -6 -4 -2 0 2 4 -4 -2 0 2 4 6 All rights reserved. Solve x ⫹ 5 ⱖ 9. 6 Solve each inequality. Graph the solution. 2. ⫺4 ⫹ w ⱕ 0 3. 3 ⫹ a ⱖ 8 4. w ⫹ 1 ⱕ 4 -6 -6 -6 -4 -4 -2 -2 0 0 2 2 4 4 6 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1. 2 ⫹ a ⬎ 6 5. y ⫹ 3 ⬍ 5 6. 6 ⫹ g ⱖ 12 7. 2 ⫹ x ⬎ 7 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 8. 2 ⫹ r ⬍ 8 -6 Course 2 Lesson 3-2 -4 -2 0 2 4 6 Reteaching 000200010271713722_CH03_L03.qxd 1/30/13 5:20 PM Name Page 2 Class Reteaching 3-3 Date Solving Inequalities by Multiplying and Dividing To solve an inequality you can multiply or divide each side by the same number. However, if the number is negative, you must also reverse the direction of the inequality sign. Solve 4y 16. Graph the solution. Solve w 3 2. Graph the solution. 4y 16 y 4 Divide each side by 4. Reverse the direction of the inequality symbol. Simplify. Graph: -6 w2 3 (3)w 3 2(3) Multiply each side by 3. w 6 Simplify. All rights reserved. 24y 16 24 24 Graph: -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 Solve each inequality. Graph the solution. 2. 4w 16 3. 2r 2 −6 −4 −2 -6 -4 -2 0 0 2 2 4 4 6 6 4. a3 1 -6 -4 -2 0 2 4 6 5. 6g 6 -6 -4 -2 0 2 4 6 6. 3x 6 -6 -4 -2 0 2 4 6 m 0 7. 22 -6 -4 -2 0 2 4 6 Course 2 Lesson 3-3 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1. 2a 10 Reteaching 000200010271713722_CH03_L04.qxd 2/13/13 3:23 PM Name Page 92 Class Reteaching 3-4 Date Solving Two-Step Inequalities You can solve a two-step inequality by using inverse operations and the properties of inequality to get the variable alone on one side of the inequality. For many inequalities, first you undo the addition or subtraction, then you undo the multiplication or division—just like when you solve a two-step equation. Undoing Addition First x 1 4 $ 28 2 x 1 4 2 4 $ 28 2 4 2 x $ 212 2 2 # x $ 2 # (212) 2 ; Simplify. ; Multiply each side by 2. ; Simplify. x $ 224 –30 –28 –26 All rights reserved. ; Subtract 4 from each side. –24 –22 –20 Undoing Subtraction First 22.4y 2 7 , 216.6 22.4y , 29.6 ; Simplify. 22.4y . 29.6 22.4 22.4 ; Divide each side by ⴚ2.4. ; Simplify. y.4 –5 –4 –3 –2 –1 0 1 2 3 4 5 Solve each inequality and graph the solution. 1. 2a 1 4 . 12 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 2. 23r 2 8 . 4 Course 2 Lesson 3-4 Reteaching © Pearson Education, Inc., publishing as Pearson Prentice Hall. 22.4y 2 7 1 7 , 216.6 1 7 ; Add 7 to each side. 000200010271713722_CH03_L04.qxd 2/13/13 3:23 PM Page 93 Name Reteaching 3-4 3. 21 n 1 8 , 9 3 Class (continued) Date Solving Two-Step Inequalities –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 –5 –4 –3 –2 –1 0 1 2 3 4 5 All rights reserved. 4. 6.25s 1 6 # 18.5 5. m 2 3 , 22 4 6. 25.6q 2 3.2 # 28.8 © Pearson Education, Inc., publishing as Pearson Prentice Hall. 7. 24 x 1 3 # 1 4 7 7 7 Reteaching Course 2 Lesson 3-4