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

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 = – x = 14.4 = 21.1 – t 2. r = r = x= = 5.4 p= 46.44

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

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

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 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

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

Pre-Algebra 3-7 Solving Inequalities with Rational Numbers A. 0.6y  1.8 y  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 – < Try This: Examples 1 Solve.

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

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   – 4949 – n   –4 Rewrite –2 as the improper fraction – Change  to . Solve. B. Multiply both sides by –. 4949

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

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

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

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:   length height

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   l 3.8

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 ≤ ≤ l and l ≤ Additional Example 3 Continued 1.3 ≤ and ≤ 2.5 l 3.8

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

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

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:   length height

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   l 4.9 Try This: Example 3 Continued

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

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

Pre-Algebra 3-7 Solving Inequalities with Rational Numbers Solve. 1. w  f – 1  x > w ≤ h 0.7 < 5.5 x > 14.2 h < f ≥ 5 Lesson Quiz: Part 1

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 miles