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6-4A Solving Compound Inequalities Involving “AND” Algebra 1 Glencoe McGraw-Hill Linda Stamper What is a compound sentence in your English class? This weekend I will go shopping and I will go to a movie. This weekend I will go shopping or I will go to a movie. A compound inequality consists of two inequalities connected by the word “AND” or the word “OR”. Today we will work with the type connected by the word “AND”. Compound inequalities involving “AND” consist of two inequalities. Both inequalities must be satisfied to make the compound inequality true. If you want to go to the movies with your friends you must finish your homework and clean your room. Writing Compound Inequalities with “AND”. Write a compound inequality that represents the set of all real numbers greater than or equal to 0 and less than 4. Then graph the inequality. n4 and n0 Compound inequalities with 0 n 4 “AND” are generally O combined in a single 4 0 inequality with the variable in the middle. • Both inequalities must be satisfied to make the compound inequality true. Therefore the solution is the intersection (overlap) of the two rays. The graph is a line segment! The number of solutions are limited. Write a compound inequality that represents the set of all real numbers greater than or equal to 2 and less than 8. Then graph the inequality. n8 and n2 Write the two inequalities. 2n 8 Rewrite as a single inequality with the O 8 2 variable in the middle. Graph. • Note: A compound inequality is usually written in a way to reflect the order of numbers on a number line. Example 1 Write a compound inequality that represents the set of all real numbers greater than or equal to -12 and less than -5. Then graph the inequality. Example 2 Write a compound inequality that represents the set of all real numbers greater than –2 and less than or equal to 3. Then graph the inequality. Example 3 Magic Mountains newest roller coaster ride has weight restrictions of greater than or equal to 50 pounds and less than 250 pounds.Write a compound inequality to describe the weight restriction and then graph the inequality. Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality. Example 1 Write a compound inequality that represents the set of all real numbers greater than or equal to -12 and less than -5. Then graph the inequality. n 5 n 12 and 12 n 5 • -12 O -5 Note: A compound inequality is usually written in a In a compound inequality involving AND, both inequalities way the of numbers a numbertrue. line. mustto bereflect satisfied toorder make the compoundoninequality Therefore the solution is the intersection (overlap) of the two rays. The graph is a line segment! The number of solutions are limited. Example 2 Write a compound inequality that represents the set of all real numbers greater than –2 and less than or equal to 3. Then graph the inequality. n 2 and n 3 2 n 3 O –2 • 3 You must label the endpoints! Example 3 Magic Mountains newest roller coaster ride has weight restrictions of greater than or equal to 50 pounds and less than 250 pounds.Write a compound inequality to describe the weight restriction and then graph the inequality. w 50 and w 250 50 w 250 • 50 O 250 Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality. Example 4 Write a compound inequality to describe when water is liquid the temperature is greater than 32 degrees F and less than 212 degrees F. Then graph the inequality. d 32 and d 212 32 d 212 O 32 O 212 Write an inequality that describes the graph. 8 x 3 • –8 O 3 1. Identify the endpoints. 2. Name the variable. 3. Write the inequality sign for each endpoint. Note: An inequality sign points toward the smaller value – thus both inequality signs point in the same direction - to the left. Example 5 Write an inequality that describes the graph. 14 x 2 • –14 • –2 1. Identify the endpoints. 2. Name the variable. 3. Write the inequality sign for each endpoint. Solving Compound Inequalities with “AND”. To solve compound inequalities with “and”, isolate the variable. You must perform the operation on all three expressions. 2 x 2 4 2 2 2 4x2 • –4 O 2 Solve. Then graph the solution. Example 6 5 2x 3 7 Example 7 3 1 2x 5 Example 8 2 2x 1 7 Example 6 Solve – 5 < 2x + 3 < 7. Then graph the solution. 5 2x 3 7 3 3 3 8 2x 4 2 2 2 4x2 • –4 O 2 Example 7 Solve –3 < –1 – 2x < 5. Then graph the solution. Reverse both inequality symbols. 3 1 2x 5 1 1 1 2 / 2x / 6 2> 2 > 2 1 x 3 3 x 1 • –3 O 1 Example 8 Solve –2 < –2x + 1 < 7. Then graph the solution. 2 2x 1 7 1 1 1 3 Reverse both / 2x / 6 inequality symbols. 2 > 2 > 2 3 x 3 2 3 3 x 2 • –3 O 3 2 To solve a compound inequality involving “AND”, isolate the variable (in the center). To perform any operation on a compound inequality involving “AND”, you must perform the operation on all three expressions. The graph of the solutions to a compound inequality involving “AND” is a line segment. 6-A5 Handout A5 (Study Guide and Intervention Page 27).