Section 1.9: Operations with Fractions, Decimals and Percent Chapter 1: Introduction to Algebra Properties of real numbers 1) Rewrite using the associative property of addition: (𝑥 + 2) + 𝑦 2) Rewrite using the associative property of addition: (𝑦 + 2) + 𝑥 Properties of real numbers 3) Rewrite using the associative property of multiplication: 6(𝑐 × 𝑑) 4) Rewrite using the associative property of multiplication: 7(𝑎 × 𝑏) Properties of real numbers 5) Rewrite using the commutative property of addition: 𝑥 + 5 6) Rewrite using the commutative property of addition: 𝑦 + 9 Properties of real numbers 7) Rewrite using the commutative property of multiplication: 𝑥5 8) Rewrite using the commutative property of multiplication: 𝑦9 Properties of real numbers 9) Rewrite using the commutative property of multiplication: (𝑥 – 3)7 10) Rewrite using the commutative property of multiplication: (𝑥 – 2)5 11) Rewrite using the distributive property, and simplify: 5(2 + 4) 12) Rewrite using the distributive property, and simplify: 7(5 + 3) 13) Rewrite using the distributive property, and simplify: (7 + 3)8 14) Rewrite using the distributive property, and simplify: (5 + 2)9 15) Rewrite using the distributive property, and simplify: 5(10 − 4) 16) Rewrite using the distributive property, and simplify: 7(5 − 3) 17) Rewrite using the distributive property, and simplify: (2 − 3)8 18) Rewrite using the distributive property, and simplify: (1 − 2)9 19) Rewrite using the distributive property: 5(𝑥 + 𝑦) 20) Rewrite using the distributive property: 7(𝑎 + 𝑏) 21) Rewrite using the distributive property: (𝑐 + 𝑑)8 22) Rewrite using the distributive property: (𝑥 + 𝑦)9 23) Rewrite using the distributive property, and simplify: 5(𝑎 − 4) 24) Rewrite using the distributive property, and simplify: 7(𝑐 − 5) 25) Rewrite using the distributive property, and simplify: (𝑥 − 2)8 26) Rewrite using the distributive property, and simplify: (𝑦 − 2)9 #27 – 38: Simplify 27) −5(𝑥 − 2) 28) −3(𝑦 – 4) 29) −(3𝑥 – 2𝑦) 30) −(5𝑥 – 4𝑦) 31) 8(3𝑥 2 + 5𝑥 – 4) 33) −2(4𝑥 2 + 6𝑥 – 3) 32) 6(2𝑦 2 + 3𝑦 – 9) 34) −3(5𝑦 3 – 6𝑦 + 1) 35) 37) 1 4 9 ( 𝑥 + 2) 2 5 3 4 6 ( 𝑥 − 5) 4 9 36) 38) 2 6 ( 𝑥+ 3 9 3 15 15 8 7 ) ( 𝑥 − 6) 5 9 Section 1.9: Operations with Fractions, Decimals and Percent Chapter 1: Introduction to Algebra. 0 Fractions in the form and division problems of the form 0 ÷ 𝑥 𝑥 Every fraction obeys the following rules 𝒏𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓 • 𝒅𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓 = 𝒂𝒏𝒔𝒘𝒆𝒓 • 𝒂𝒏𝒔𝒘𝒆𝒓 × 𝒅𝒆𝒏𝒐𝒎𝒊𝒏𝒂𝒕𝒐𝒓 = 𝒏𝒖𝒎𝒆𝒓𝒂𝒕𝒐𝒓 For example: 6 2 =3 and 3× 2 = 6 0 What does the fraction 6 𝑟𝑒𝑑𝑢𝑐𝑒 𝑡𝑜? I know that the above rule must apply to the answer: 0 6 = 𝑎𝑛𝑠𝑤𝑒𝑟 and 𝑎𝑛𝑠𝑤𝑒𝑟 × 6 = 0 0 I claim: 6 = 0 I will show you that it does. 0 𝑤𝑒 𝑘𝑛𝑜𝑤 𝑡ℎ𝑎𝑡 6 = 𝑎𝑛𝑠𝑤𝑒𝑟 • • 𝑐ℎ𝑎𝑛𝑔𝑒 𝑎𝑛𝑠𝑤𝑒𝑟 𝑡𝑜 0 0 = 0 Then: 0× 6 = 0 6 • Thus 𝑎𝑛𝑑 𝑎𝑛𝑠𝑤𝑒𝑟 × 6 = 0 𝑠𝑖𝑛𝑐𝑒 0 × 6 𝑑𝑜𝑒𝑠 𝑒𝑞𝑢𝑎𝑙 0 𝐼 𝑘𝑛𝑜𝑤 𝑚𝑦 𝑎𝑛𝑠𝑤𝑒𝑟 𝑖𝑠 𝑐𝑜𝑟𝑟𝑒𝑐𝑡 0 6 =0 Rule for fractions with 0 in the numerator 0 𝑥 = 0; 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑥 ≠ 0 and 0 ÷ 𝑥 = 0; 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑥 ≠ 0 (since 0 𝑥 = 0 ÷ 𝑥) Section 1.9: Operations with Fractions, Decimals and Percent Chapter 1: Introduction to Algebra. Fractions in the form 𝑥 0 , (𝑥 ≠ 0) and division problems in the form 𝑥 ÷ 0, 𝑥 ≠ 0 12 What does the fraction 0 𝑟𝑒𝑑𝑢𝑐𝑒 𝑡𝑜? I know that the above rule must apply to the answer: 12 0 = 𝑎𝑛𝑠𝑤𝑒𝑟 and 𝑎𝑛𝑠𝑤𝑒𝑟 × 0 = 12 This is a problem when we try to figure out how to solve this: 𝑎𝑛𝑠𝑤𝑒𝑟 × 0 = 12 • The left side will always reduce to zero, no matter what number I replace with the word answer. The times 0 on the left side forces the left side to always equal 0. • There is no number to change the word answer to so that the left side will equal 12 12 • There is no way to reduce the fraction 0 Thus, we say 𝑥 0 12 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 Rule for fractions with 0 in the denominator: = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑; 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑥 ≠ 0 and x ÷ 0 = 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑; 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 𝑥 ≠ 0 (since 𝑥 0 = x ÷ 0) #39 – 46: Simplify 39) 5 0 41) 12 ÷ 0 43) 0 7 45) 0 ÷ 3 40) 2 0 42) 8 ÷ 0 44) 0 6 46) 0 ÷ 9