1. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers 3-7 Solving Inequalities with Rational Numbers Pre-Algebra Warm Up Warm Up Problem of the Day Problem of the Day Lesson Presentation Lesson Presentation

2. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers p 8.6 Warm Up Solve. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers 1. t + 8.7 = – 12.4 3. 3.2x = 14.4 = 21.1 – t 2. r + 1313 11 = 4 4949 r = 6 8989 x= 4.5 4. = 5.4 p= 46.44

3. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Problem of the Day Arnie built a fence around a rectangular field that measures 120 feet by 90 feet. He put a post in each corner and every 6 feet along all four sides. How many fence posts did he use? 70 posts

4. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Learn to solve inequalities with rational numbers.

5. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers The minimum size for a piece of first-class mail is 5 inches long, 3 inches wide, and 0.007 inch thick. For a piece of mail, the combined length of the longest side and the distance around the thickest part may not exceed 108 inches. Many inequalities are used in determining postal rates. 1212

6. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers A. 0.4x  0.8 Solve. x  2 0.8 0.40.4 x  Divide both sides by 0.4. y < 14.8 B. y – 3.8 < 11 y – 3.8 + 3.8 < 11 + 3.8 Additional Examples 1: Solving Inequalities with Decimals Add 3.8 to both sides of the equation.

7. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers A. 0.6y  1.8 y  3 1.8 0.60.6 y  Divide both sides by 0.6. Add 4.2 to both sides of the equation. m < 19.2 B. m – 4.2 < 15 m – 4.2 + 4.2 < 15 + 4.2 Try This: Examples 1 Solve.

8. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers x +  2 2323 Subtract from both sides. 2323 x + –  2 – 2323 2323 2323 x >  1 1313 Additional Example 2A: Solving Inequalities with Fractions Solve. A.

9. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers –2 n   9 Additional Example 2B: Solving Inequalities with Fractions Continued 1414 When multiplying or dividing an inequality by a negative number, reverse the inequality symbol. Remember! – n   9 9494 9494 4949 – 4949 – n   –4 Rewrite –2 as the improper fraction –. 1414 9494 Change  to . Solve. B. Multiply both sides by –. 4949

10. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers v +  7 1515 v + –  7 – 1515 1515 1515 v >  6 4545 Try This: Example 2A Solve. A. Subtract from both sides. 1515

11. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers –1 j ≤  7 2525 – j ≤  7 7575 7575 5757 – 5757 – j ≥  –5 Try This: Example 2B Solve. B. Rewrite –1 as the improper fraction –. 2525 7575 Change  to . Multiply both sides by –. 5757

12. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Additional Example 3: Problem Solving Application The height of an envelope is 3.8 in. What are the minimum and maximum lengths to avoid an extra charge? With first-class mail, there is an extra charge in any of these cases: The length is greater than 11 inches The height is greater than 6 inches The thickness is greater than inch The length divided by the height is less than 1.3 or greater than 2.5 1212 1818 1414

13. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers The answer is the minimum and maximum lengths for an envelope to avoid an extra charge. List the important information: The height of the piece of mail is 3.8 inches. If the length divided by the height is between 1.3 and 2.5, there will not be an extra charge. Understand the Problem Additional Example 3 Continued Show the relationship of the information: 1.3 2.5   length height

14. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Make a Plan You can use the model from the previous slide to write an inequality where l is the length and 3.8 is the height. Additional Example 3 Continued 1.3 2.5   l 3.8

15. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers The length of the envelope must be between 4.94 in. and 9.5 in. Solve Multiply both sides of each inequality by 3.8. Simplify. l  4.94 and l ≤ 9.5 3.8 1.3 ≤ l and l ≤ 2.5 3.8 Additional Example 3 Continued 1.3 ≤ and ≤ 2.5 l 3.8

16. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers 4.94  3.8 = 1.3 and 9.5  3.8 = 2.5, so there will not be an extra charge. Look Back Additional Example 3 Continued

17. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Try This: Example 3 The height of an envelope is 4.9 in. What are the minimum and maximum lengths to avoid an extra charge? With first-class mail, there is an extra charge in any of these cases: The length is greater than 11 inches The height is greater than 6 inches The thickness is greater than inch The length divided by the height is less than 1.3 or greater than 2.5 1212 1818 1414

18. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers The answer is the minimum and maximum lengths for an envelope to avoid an extra charge. List the important information: The height of the piece of mail is 4.9 inches. If the length divided by the height is between 1.3 and 2.5, there will not be an extra charge. Understand the Problem Try This: Example 3 Continued Show the relationship of the information: 1.3 2.5   length height

19. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Make a Plan You can use the model from the previous slide to write an inequality where l is the length and 4.9 is the height. 1.3 2.5   l 4.9 Try This: Example 3 Continued

20. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers The length of the envelope must be between 6.37 in. and 12.25 in. Solve Multiply both sides of each inequality by 4.9. Simplify. l  6.37 and l ≤ 12.25 4.9 1.3 ≤ l and l ≤ 2.5 4.9 1.3 ≤ and ≤ 2.5 l 4.9 Try This: Example 3 Continued

21. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers 6.37  4.9 = 1.3 and 12.25  4.9 = 2.5, so there will not be an extra charge. Look Back Try This: Example 3 Continued

22. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Solve. 1. w + 1.25  5.12 2. f – 1  4 3. 1.2x > 17.04 4. w ≤ 3.87 3838 1414 h 0.7 < 5.5 x > 14.2 h < 3.85 5858 f ≥ 5 Lesson Quiz: Part 1

23. Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Rosa’s car gets between 20 and 21 mi/gal on the highway. She knows that her gas tank holds at least 18 gallons. Using a full tank of gas, what is the minimum distance Rosa could drive her car on the highway between fill-ups? Lesson Quiz: Part 2 5. 360 miles