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Warm Up Solve each inequality. 1. x + 3 ≤ 10 2.

Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. . x ≤ 7. 23 < –2 x + 3 . –10 > x. Solve each inequality and graph the solutions. 4. 4 x + 1 ≤ 25 . x ≤ 6. 5. 0 ≥ 3 x + 3. –1 ≥ x. Section 3.6. Solving Compound Inequalities. California Standards.

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Warm Up Solve each inequality. 1. x + 3 ≤ 10 2.

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  1. Warm Up Solve each inequality. 1. x + 3 ≤ 10 2. x ≤7 23 < –2x + 3 –10> x Solve each inequality and graph the solutions. 4. 4x + 1 ≤ 25 x ≤ 6 5. 0 ≥ 3x + 3 –1 ≥ x

  2. Section 3.6 Solving Compound Inequalities

  3. California Standards 5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

  4. Vocabulary The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.

  5. < Less than Reviewing Inequalities Symbols > Greater than ≤ Less than or equal to ≥ Greater than or equal to

  6. Identify each symbol • ≥ • < • = • ≤ • > • ≠ • ≈ “is greater than or equal to” “is less than” “is equal to” “ is less than or equal to” “ is greater than” “ is not equal to” “is approximately equal to”

  7. Let’s try one together. 4<x+2<8 Step 1: Make two equations Step 2: undo addition or subtraction Step 3: Solve Step 4: Graph 4<x+2 x+2<8 -2 -2 -2 -2 2<x x<6 2 < x < 6 0 2 4 6 8 All real numbers greater than or equal to two AND all real numbers less than six.

  8. Let’s try one together. -5<2x+3<9 Step 1: Make two equations Step 2: undo addition or subtraction Step 3: undo division and multiplication Step 4: Solve Step 5: Graph -5<2x+3 2x+3<9 -3 -3 -3 -3 -8<2x 2x<6 2 2 -4 < x < 3 2 2 All real numbers greater than or equal to negative four AND all real numbers less than three. -4 -2 0 2 4

  9. 8 < 3x – 1 ≤ 11 +1 +1 +1 9 < 3x ≤ 12 Short Method Solve the compound inequality and graph the solutions. 8 < 3x – 1 ≤ 11 1. add 1 to each part of the inequality. 2. divide each part of the inequality by 3 to undo the multiplication. 3 < x ≤ 4 The solution set is {x:3 < x ≤ 4}.

  10. 3 < x ≤ 4 Graph 3 < x. Graph x ≤ 4. Graph the intersection by finding where the two graphs overlap. –3 –2 0 1 2 3 4 5 –4 –1 –5

  11. Let’s try one together. -4 + a > 1 OR -4 + a < -3 Step 1: It’s already two equations Step 2: undo addition or subtraction Step 3: Solve Step 4: Graph -4 + a > 1 -4 + a < -3 +4 +4 +4 +4 a > 5 a<1 a > 5 OR a < 1 -1 1 3 5 7 All real numbers greater than five OR all real numbers less than one.

  12. Let’s try one together. r – 2 < 0 OR r – 1 > 4 Step 1: It’s already two equations Step 2: undo addition or subtraction Step 3: Solve Step 4: Graph r – 2 < 0 r – 1 > 4 +2 +2 +1 +1 r < 2 r > 5 -1 1 3 5 7 a < 2 OR a > 5 All real numbers greater than five OR all real numbers less than two.

  13. Match the Compound Inequality with the Correct Graph • 0 < x + 2 < 5 • -4 + a > 1 OR -4 + a < -3 • -3 < x + 2 < 3 • 2 < x + 2 < 5 • x + 2 < -6 OR x + 2 > -2 -8 -6 -4 -4 -2 0 -2 0 2 1 3 5 0 2 4

  14. Now You Try…Solve and Graph the Compound Inequality • -3 < x + 2 < 7 • x – 1 < -1 OR x – 5 > -1 • 2 < x + 2 < 5 • 11 < 2x + 3 < 21 • n + 2 < 3 OR n + 3 > 7 -5 0 5 -5 < x < 5 0 2 4 x < 0 OR x > 4 0 2 4 0 < x < 3 4 6 8 4 < x < 9 x < 1 OR x > 4 1 3 5

  15. Write the compound inequality shown by the graph. The compound inequality is x ≤ –8 OR x > 0.

  16. Write the compound inequality shown by the graph. The compound inequality is –9 < x AND x < –2 (or –9 < x < –2).

  17. Match the Following Inequality Natural Numbers Inverse Operations Like Terms Compound Inequality the set of counting numbers two inequalities that are combined into one statement by the word AND or OR. terms that contain the same variable raised to the same power a mathematical statement that compares two expressions by using one of the following signs: <, >, <, >, or ≠ operation that “undo” each other

  18. In Summary • Today you learned that two inequalities that are combined into one statement by the word AND or OR is called a compound inequality. • If it contains the word AND it is split into two equations and the graph is in between two points. • If it contains the word OR the graphs go in opposite directions from each point.

  19. Lesson Quiz Solve each compound inequality and graph the solutions. 1. 2 ≤ 2w + 4 ≤ 12 –1 ≤ w ≤ 4 2. 3 + r > −2 OR 3 + r < −7 r > –5 OR r < –10 Write the compound inequality shown by each graph. 4. x < −7 OR x ≥ 0 5. −2 ≤ a < 4

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