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6.3 Solving Compound Inequalities

Objective 1. Objective 2. 6.3 Solving Compound Inequalities. Model a real-life situation with a compound inequality. Write, solve and graph compound inequalities. VOCABULARY A compound inequality consists of two inequalities connected by the word and or the word or .

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6.3 Solving Compound Inequalities

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  1. Objective 1 Objective 2 6.3 Solving Compound Inequalities Model a real-life situation with a compound inequality. Write, solve and graph compound inequalities. VOCABULARY A compound inequality consists of two inequalities connected by the word and or the word or.

  2. EX: Graph x > –2 and x < 3 X > -2 X < 3 -3 -2 -1 0 1 2 3 NOTE: Solutions are all numbers that are greater than or equal to –2 and less than 3. Since X is between –2 and 3 the inequality is usually written as –2 < X < 3 EX: Graph X < 3orX > 6 Graph both equations on the same graph 1 2 3 4 5 6 7

  3. Example 2 Compound Inequalities in Real Life Write an inequality that describes each condition. a. Water is a non-liquid when the temperature is 32°F or below or is at least 212°F. T ≤ 32 or T ≥ 212 b. A refrigerator is designed to work on an electric line carrying from 115 volts to 120 volts. 115 ≤ V ≤ 120

  4. -5 -4 -3 -2 -1 0 1 2 3 Example 3 Solving a Compound Inequality with And Solve -5 ≤ 2x + 3 < 7. Then graph the solution. Isolate the variable between the inequality symbols. –5 ≤ 2x + 3 < 7 -3 -3 -3 Subtract 3 from all three sides –8 ≤ 2x < 4 2 2 2 Divide each side by 2. –4 ≤ x < 2 The solution is all real numbers that are greater than or equal to -4 and less than 2.

  5. Solving a Compound Inequality with Or Solve x + 5 < -6 or 3x > 12 Solve each part separately. 3x >12 or x + 5 ≤ –6 3 3 -5 -5 or x ≤ –11 x > 4 -12 -10 -8 -6 -4 -2 0 2 4 6

  6. Example 5 Reversing the sign Solve –3 < –1 – 2x ≤ 5. Then graph the solution. +1 +1 +1 –2 < – 2x ≤ 6 Reverse the inequalities when you divide by a negative –2 –2 –2 1 > x ≥ –3 -4 -3 -2 -1 0 1 2

  7. -10 -9 -8 -7 -6 -5 -4 -3 -2 -4 -3 -2 -1 0 1 2 3 4 -2 -1 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 6 Try these Solve and graph the inequality. 1. 2. 3. 4.

  8. Modeling with a Compound Inequality The park is 2 miles from your house. The mall is 3 miles from your house. a. Find the minimum distance between the park and the mall. b. Find the maximum distance between the park and the mall. c. Write an inequality that describes the possible distances d between the park and the mall. Solution – DRAW A DIAGRAM • The shortest distance is 1 mile. The mall is somewhere on this circle b. The longest distance is 5 miles. 3 2 c. The values of d can be described by the inequality 1 ≤ d ≤ 5 House The park is somewhere on this circle

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