1. 6-3 Using Properties with Rational Numbers Warm Up Identify the property represented. 1. 4 + (9 + 3) = (4 + 9) + 3 2. 10(5 - 6) = 10. 5 - 10. 6 3. 17. 1 = 17 Associative Property Distributive Property Identity Property

2. 6-3 Using Properties with Rational Numbers Learn to use properties of rational numbers to write equivalent expressions and equations.

3. 6-3 Using Properties with Rational Numbers The Distributive Property states: a(b + c) = ab + ac a(b - c) = ab - ac Remember

4. 6-3 Using Properties with Rational Numbers Additional Example 1: Writing Equivalent Expressions An art teacher pays $13.89 for one box of watercolor brushes. She buys 6 boxes in March and 5 boxes in April. Use the Distributive Property to write equivalent expressions showing two ways to calculate the total cost of the watercolor boxes. Write an expression to show how much the teacher pays for a box and how many boxes purchased. Then use the Distributive Property to write an equivalent expression

5. 6-3 Using Properties with Rational Numbers Method 1 $13.89(6 + 5) $13.89(11) $152.79 Method 2 $13.89(6) + $13.89(5) $83.34 + $69.45 $152.79 Both methods result in a calculation of $152.79 for the amount of money spent of watercolor brushes.

6. 6-3 Using Properties with Rational Numbers Check It Out : Example 1 Jamie earns $8.75 per hour. Last week she worked 15 hours and next week she will work 20 hours. Use the Distributive Property to write equivalent expressions showing two ways to calculate how much money she earned. Write an expression to show how much Jamie earns and the number of hours she works. Then use the Distributive Property to write an equivalent expression.

7. 6-3 Using Properties with Rational Numbers Method 1 $8.75(15 + 20) $8.75(35) $306.25 Method 2 $8.75(15) + $8.75(20) $131.25 + $175 $306.25 Both methods result in a calculation of $306.25 for Jamie’s salary. Continued: Check It Out Example 1

8. 6-3 Using Properties with Rational Numbers Write an equivalent equation for that does not contain fractions. Then solve the equation. Check It Out: Example 2 1212 X + 9 = 4646 1212 4646 The LCM of denominators is 6 1212 x+ 9 = 4646 66 Multiply both sides by 6. 1 1 2 x + 6 (9) = 4 1 6 3 6 6 1 Simplify.

9. 6-3 Using Properties with Rational Numbers 3x + 54 = 4 3x + 54 = 4 is an equivalent expression 3x + 54 = 4 -54 3x = -50 33 Subtract 54 from both sides. Divide both side by 3 x = -16 2323 An equivalent equation is 3x + 34 = 4 and the solution is x = -16 2323 Continued: Check It Out Example 2

10. 6-3 Using Properties with Rational Numbers The soccer team uses a 36.75-liter container to take water to games. The team manager fills 0.75 liter bottles from this. He has used 22.5 liters. How many more 0.75 liter bottles can he fill before he runs out of water? Write and solve an equivalent equation without decimals. Additional Example 3: Construction Application Write an equation to represent the situation. 0.75x + 22.5 = 36.75

11. 6-3 Using Properties with Rational Numbers Write an equivalent equation without decimals. 100(0.75x + 22.5) = (36.75)100 The equation has decimals to the hundredths, so multiply both sides by 100. Use the Distributive Property 100(0.75x + 100(22.5) = (36.75)100 75x + 2,250 = 3,675 Simplify to get an equivalent equation without decimals Continued: Example 3

12. 6-3 Using Properties with Rational Numbers 75x + 2,250 = 3,675 -2250 75x = 1,425 75 x = 19 The number of 0.75 liter bottles that he can fill before he runs out of water is 19. Continued: Example 3

13. 6-3 Using Properties with Rational Numbers Check It Out: Example 3 …If the soccer team uses a 42.5-liter container, about how many 0.75 liter bottles can the manager fill before he runs out of water? Write an equation to represent the situation. 0.75x + 22.5 = 42.5 Write an equivalent equation without decimals. 100(0.75x + 22.5) = (42.5)100 The equation has decimals to the hundredths, so multiply both sides by 100.

14. 6-3 Using Properties with Rational Numbers Use the Distributive Property 100(0.75x + 100(22.5) = (42.5)100 75x + 2,250 = 4,250 Simplify to get an equivalent equation without decimals Continued: Check It Out Example 3 75x + 2,250 = 4,250 -2250 75x = 2000 75

15. 6-3 Using Properties with Rational Numbers Continued: Check It Out Example 3 The number of 0.75 liter bottles that he can fill before he runs out of water is 19. x ≈ 26.6

16. 6-3 Using Properties with Rational Numbers Lesson Quiz 1. Jai earns $9.75 per hour. Jai works 3 hours one day and then works 7 hours the next day. Use the Distributive Property to write equivalent expressions showing two ways to calculate Jai’s total earnings. 9.75(3) + 9.75(7); 9.75(3 + 7); $97.50 Write an equivalent equation that does not contain fractions. Then solve the equation. 4545 x + 4 = 1212 2. 8x + 40 = 5;x = -4 3838

17. 6-3 Using Properties with Rational Numbers Lesson Quiz 2323 x - 4 = 1414 3. 8x - 48 = 3;x = 6 3838 4. Joy has $67.85. She buys several pairs of earrings at $9.98 per pair and has $17.95 left. How many pairs of earrings did she buy? Write and solve an equivalent equation without decimals. 9.98x + 17.95 = 67.85; 998x + 1795 = 6785; x = 5; Joy bought 5 pairs of earrings.