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Graphing Systems of Linear Inequalities: Examples and Practice

This lesson provides a comprehensive introduction to graphing systems of linear inequalities. Students will learn how to graph inequalities using color coding to identify regions of solutions. Through engaging examples, they will explore various scenarios, including shopping situations that lead to systems of inequalities. After walking through step-by-step graphing methods, students will analyze and interpret the common shaded regions that represent solutions. Practice exercises reinforce learning and application, ensuring students grasp the concepts effectively.

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Graphing Systems of Linear Inequalities: Examples and Practice

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  1. Graph Systems of Linear Inequalities Warm Up Lesson Presentation Lesson Quiz

  2. ANSWER ANSWER Warm-Up 1.Graphy ≤ x + 2. 2.Graphy ≥ | x |.

  3. Example 1 Graph the system of inequalities. y > –2x – 5 Inequality 1 y <x + 3 Inequality 2

  4. STEP 1 Graph each inequality in the system. Use red for y > –2x – 5and bluefory≤ x + 3. STEP 2 Identify the region that is common to both graphs. It is the region that is shaded purple. Example 1 SOLUTION

  5. 2 y > – x + 4 3 Example 2 Graph the system of inequalities. 2x + 3y < 6 Inequality 1 Inequality 2

  6. STEP 1 Graph each inequality in the system. Use red for2x + 3y <6and bluefory > – x + 4. 2 3 STEP 2 Identify the region that is common to both graphs. There is no region shaded both red and blue. So, the system has no solution. Example 2 SOLUTION

  7. y > x + 4 Example 3 Graph the system of inequalities. y < 3 Inequality 1 Inequality 2

  8. STEP 1 Graph each inequality in the system. Use red for y ≤3and bluefor y > x + 4 . STEP 2 Identify the region that is common to both graphs. It is the region that is shaded purple. Example 3 SOLUTION

  9. 1 2.2x – y > 4 2 4x – y < 5 Guided Practice Graph the system of inequalities. 1. y < 3x – 2 y > – x + 4

  10. 4. y < 4 3. x + y > – 3 y >x – 5 –6x + y < 1 Guided Practice

  11. Example 4 SHOPPING A discount shoe store is having a sale, as described in the advertisement shown. • Use the information in the ad to write a system of inequalities for the regular footwear prices and possible sale prices. • Graph the system of inequalities. • Use the graph to estimate the range of possible sale prices for footwear that is regularly priced at $70.

  12. STEP 1 Write a system of inequalities. Let xbe the regular footwear price and let ybe the sale price. From the information in the ad, you can write the following four inequalities. Example 4 SOLUTION x ≥ 20 Regular price must be at least $20. x ≤ 80 Regular price can be at most $80. y ≥ 0.4x Sale price is at least (100 – 60)% = 40% of regular price. y ≤ 0.9x Sale price is at most (100 – 10)% = 90% of regular price.

  13. Example 4 STEP 2 Graph each inequality in the system. Then identify the region that is common to all the graphs. It is the region that is shaded. STEP 3 Identify the range of possible sale prices for $70 footwear. From the graph you can see that when x = 70, the value of yis between these values: 0.4(70) = 28 and 0.9(70) = 63 So, the value ofysatisfies28 ≤ y ≤ 63. ANSWER Therefore, footwear regularly priced at $70 sells for between $28 and $63, inclusive, during the sale.

  14. 7. What if? In Example 4, suppose the advertisement showed a range of discounts of 20% – 50% and a range of regular prices of $40 – $100. Guided Practice a. Write and graph a system of inequalities for the regular footwear prices and possible sale prices. x ≥ 40 Regular price must be at least $40. x ≤ 100 Regular price can be at most $100. y ≥ 0.5x Sale price is at least (100 – 50)% = 50% of regular price. y ≤ 0.8x Sale price is at most (100 – 20)% = 80% of regular price.

  15. ANSWER 30≤y≤ 48 b. Use the graph to estimate the range of possible sale prices for footwear that is regularly priced at $60. Guided Practice

  16. ANSWER Lesson Quiz Graph the system of inequalities. 1. y>–x – 4 y< 2x + 2

  17. 2. 2x – 4y< 8 y < – x + 4 ANSWER 1 3 Lesson Quiz Graph the system of inequalities.

  18. 3. y > x + 2 y< 4 x < – 1 ANSWER Lesson Quiz Graph the system of inequalities.

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