Mathematics and Statistics PDF (math114)

Mathematics and Statistics (math114) College of Business Administration Jazan University 1 __________ math114 Chapter R: Basic Algebraic Operations R-1 Algebra and Real Numbers Ø The Set of Real Numbers: Ø The Real Number Line: Ø...

Mathematics and Statistics (math114) College of Business Administration Jazan University 1 __________ math114 Chapter R: Basic Algebraic Operations R-1 Algebra and Real Numbers Ø The Set of Real Numbers: Ø The Real Number Line: Ø Addition and Multiplication of Real Numbers: Ex. Perform the indicated operations: 𝟏 𝟔 (𝟏)(𝟓)'(𝟑)(𝟔) 𝟓'𝟏𝟖 𝟐𝟑 A. + = 𝟑 𝟓 = = (𝟑)(𝟓) 𝟏𝟓 𝟏𝟓 𝟏 𝟏 B. + = 𝟑 𝟓 𝟑 𝟒 C. + = 𝟒 𝟑 2 __________ math114  𝟖 𝟓 (𝟖)(𝟓) 𝟒𝟎 𝟏𝟎 ,-.,∙0- 0- D. ∙ = (𝟑)(𝟒) = = 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 = 𝟑 𝟒 𝟏𝟐 𝟑 01.,.3 3 𝟐 𝟒 E.. = 𝟑 𝟕 𝟑 𝟓 F. ,−. ,−. = 𝟓 𝟑 Ø Further Operations and Properties § Basic Properties of The Set of Real Numbers: Addition Properties & Multiplication Properties: These operations are 𝟑 𝟓 𝟓 𝟑 Commutative (+): 𝒙 + 𝒚 = 𝒚 + 𝒙 𝒆𝒙. + = + 𝟐 𝟕 𝟕 𝟐 𝟑 𝟓 𝟓 𝟑 Commutative (.): 𝒙𝒚 = 𝒚𝒙 𝒆𝒙.. =. 𝟐 𝟕 𝟕 𝟐 𝟑 𝟓 𝟗 𝟑 𝟓 𝟗 Associative (+): (𝒙 + 𝒚) + 𝒛 = 𝒙 + (𝒚 + 𝒛) 𝒆𝒙. +, +. =, +.+ 𝟐 𝟕 𝟒 𝟐 𝟕 𝟒 𝟑 𝟓 𝟗 𝟑 𝟓 𝟗 Associative (.): (𝒙. 𝒚). 𝒛 = 𝒙. (𝒚. 𝒛) 𝒆𝒙.. ,.. = , 𝟐. 𝟕.. 𝟒 𝟐 𝟕 𝟒 𝟓 𝟓 𝟓 𝟎 𝒊𝒔 𝒂𝒏 𝒂𝒅𝒅𝒊𝒕𝒊𝒗𝒆 𝒊𝒅𝒆𝒏𝒕𝒊𝒕𝒚 (+): 𝟎 + 𝒙 = 𝒙 + 𝟎 = 𝒙 𝒆𝒙. 𝟎 + 𝟐 = 𝟐 + 𝟎 = 𝟐 = 𝟓 𝟓 𝟓 𝟏 𝒊𝒔 𝒂 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒕𝒊𝒗𝒆 𝒊𝒅𝒆𝒏𝒕𝒊𝒕𝒚 (. ): 𝟏. 𝒙 = 𝒙. 𝟏 = 𝒙 𝒆𝒙. 𝟏. =. 𝟏 = 𝟐 𝟐 𝟐 Additive inverse (+): 𝒇𝒐𝒓 𝒆𝒂𝒄𝒉 𝑥 𝒊𝒏 𝑹, −𝑥 𝒊𝒕𝒔 𝑎𝑑𝑑𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 ; 𝒕𝒉𝒂𝒕 𝒊𝒔 𝒙 + (−𝒙) = (−𝒙) + 𝒙 = 𝟎 𝟓 𝟓 𝟓 𝟓 𝟓 𝟓 𝒆𝒙. 𝟒 𝒊𝒕𝒔 𝒂𝒅𝒅𝒊𝒕𝒊𝒗𝒆 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 − 𝟒 ;𝒕𝒉𝒂𝒕 𝒊𝒔 𝟒 + ,− 𝟒. = 𝟒 − 𝟒 = 𝟎 Multiplicative inverse (.): 0 𝒇𝒐𝒓 𝒆𝒂𝒄𝒉 𝑥 𝒊𝒏 𝑹, 𝒙 ≠ 𝟎, 𝑥 60 = 𝒊𝒔 𝒊𝒕𝒔 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒗𝒆 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 7 6𝟏 6𝟏 ; 𝒕𝒉𝒂𝒕 𝒊𝒔 𝒙𝒙 = 𝒙 𝒙 = 𝟏 𝟓 𝟓 𝟒 𝟓 𝟒 𝟓.𝟒 𝟐𝟎 𝒆𝒙. 𝒊𝒕𝒔 𝒂 𝒎𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒄𝒂𝒕𝒊𝒗𝒆 𝒊𝒏𝒗𝒆𝒓𝒔𝒆 ( )6𝟏 = ; 𝒕𝒉𝒂𝒕 𝒊𝒔. = = =𝟏 𝟒 𝟒 𝟓 𝟒 𝟓 𝟒.𝟓 𝟐𝟎 𝟏𝟕 𝟗 𝒆𝒙. (− 𝟗 )$𝟏 = − 𝟏𝟕 Distributive: 𝒙(𝒚 + 𝒛) = 𝒙𝒚 + 𝒙𝒛 (𝒙 + 𝒚)𝒛 = 𝒙𝒛 + 𝒚𝒛 Ex. Which real number property justifies the indicated statement? A. (𝟕𝒙)𝒚 = 𝟕(𝒙𝒚) … … … … … … … …. B. 𝒂(𝒃 + 𝒄) = 𝒂𝒃 + 𝒂𝒄 ……………… C. 𝒂(𝒃 + 𝒄) = (𝒃 + 𝒄)𝒂 ……………… 3 __________ math114  D. (𝟐𝒙 + 𝟑𝒚) + 𝟓𝒚 = 𝟐𝒙 + (𝟑𝒚 + 𝟓𝒚) … … … … …. E. (𝒙 + 𝒚)(𝒂 + 𝒃) = (𝒙 + 𝒚)𝒂 + (𝒙 + 𝒚)𝒃 … … … … … … …. F. 𝑰𝒇 𝒂 + 𝒃 = 𝟎 , 𝒕𝒉𝒆𝒏 𝒃 = −𝒂 … … … … … … …. Ex. Perform the indicated operations: A. 𝟏𝟎𝟎 ÷ 𝟎 = 𝐷𝑖𝑣𝑖𝑠𝑖𝑜𝑛 𝑏𝑦 0 𝑛𝑒𝑣𝑒𝑟 𝑎𝑙𝑙𝑜𝑤𝑒𝑑 𝟏 𝟏 B. 𝟐 + 𝟕 = 𝟖 𝟒 C. − = 𝟗 𝟓 𝟏 𝟑 D. ,−.. = 𝟏𝟎 𝟖 E. 𝟎 ÷ 𝟎 = 𝟒 𝟔 F. 𝟕 ÷ ,𝟑 − 𝟐. = G. −(𝟒6𝟏 + 𝟑) = 𝟑 6𝟏 𝟖 𝟏 (𝟖)(𝟐)'(𝟑)(𝟏) 𝟏𝟔'𝟑 𝟏𝟗 H. ,. + 𝟐6𝟏 = + = = = 𝟖 𝟑 𝟐 (𝟑)(𝟐) 𝟔 𝟔 𝟗 6𝟏 6𝟔 𝟗 6𝟏 (6𝟔)(𝟐)'(𝟏)(𝟗) 6𝟏 6𝟏𝟐'𝟗 6𝟏 6𝟑 6𝟏 𝟐 I. ,−𝟔 +. =, +. =,. =,. =,. = 𝟐 𝟏 𝟐 (𝟏)(𝟐) 𝟐 𝟐 6𝟑 4 __________ math114 R-2 Exponents and Radicals Ø Integer Exponents: Ex. A. 𝟐𝟓 = 𝟐. 𝟐. 𝟐. 𝟐. 𝟐 = 𝟑𝟐 𝟏 𝟏 𝟏 B. 𝟕$𝟑 = 𝟕.𝟑 = 𝟕.𝟕.𝟕 = 𝟑𝟒𝟑 C. 𝟐𝟑𝟎 = 𝟏 Ex. Write using positive exponents or decimals: A. (𝒖𝟑 𝒗𝟐 )𝟎 = 𝟏 𝟏 B. 𝒙$𝟖 = 𝒙𝟖 𝟏 𝟏 𝟏 C. 𝟏𝟎$𝟑 = = = = 𝟎. 𝟎𝟎𝟏 𝟏𝟎𝟑 (𝟏𝟎)(𝟏𝟎)(𝟏𝟎) 𝟏𝟎𝟎𝟎 𝒙#𝟑 𝒙#𝟑 𝟏 𝟏 𝒚𝟓 𝟏(𝒚𝟓 ) 𝒚𝟓 D. 𝒚#𝟓 = 𝟏. 𝒚#𝟓 = 𝒙𝟑. 𝟏 = (𝒙𝟑 )(𝟏) = 𝒙𝟑 Ex. Simplify using exponents properties, and express answer using positive exponents only? A. c𝟑𝒂𝟓 d(𝟐𝒂6𝟑 ) = (𝟑. 𝟐)c𝒂𝟓 𝒂6𝟑 d = 𝟔𝒂𝟓'(6𝟑) = 𝟔𝒂𝟓6𝟑 = 𝟔𝒂𝟐 𝟏 𝒂𝟔 𝟏 𝟏 𝒂𝟔 𝟏 𝒂𝟔 𝒂𝟔 B. (𝟐𝒂$𝟑 𝒃𝟐 )$𝟐 = 𝟐(𝟏)($𝟐) 𝒂($𝟑)($𝟐) 𝒃(𝟐)($𝟐) = 𝟐$𝟐 𝒂𝟔 𝒃$𝟒 = 𝟐𝟐 𝟏 𝒃𝟒 = 𝟐.𝟐 𝒃𝟒 = 𝟒 𝒃𝟒 = 𝟒𝒃𝟒 𝟔𝒙!𝟐 𝟔 𝒙!𝟐 𝟑.𝟐 𝟑 𝟑 𝟑 𝒙𝟑 𝟑𝒙𝟑 C. = 𝟖. = 𝟒.𝟐. 𝒙6𝟐6(6𝟓) = 𝟒. 𝒙6𝟐'𝟓 = 𝟒 𝒙𝟑 = 𝟒. = 𝟖𝒙!𝟓 𝒙!𝟓 𝟏 𝟒 D. −𝟒𝒚𝟑 − (−𝟒𝒚)𝟑 = −𝟒𝒚𝟑 − (−𝟒)𝟑 𝒚𝟑 5 __________ math114  = −𝟒𝒚𝟑 − (−𝟒)(−𝟒)(−𝟒)𝒚𝟑 = −𝟒𝒚𝟑 − (+𝟏𝟔)(−𝟒)𝒚𝟑 = −𝟒𝒚𝟑 − (−𝟔𝟒)𝒚𝟑 = −𝟒𝒚𝟑 + 𝟔𝟒𝒚𝟑 = +𝟔𝟎𝒚𝟑 Ex. evaluate each expression. If the answer is not an integer, write it in fraction form: A. 𝟑𝟕 = 𝟏 𝟖 B. ,𝟐. = C. 𝟔6𝟑 = D. (−𝟓)𝟒 = E. (−𝟕)6𝟐 = F. −𝟏𝟎 = Ø Roots of Real Numbers: Theorem: Number of Real nth Roots of a Real Number b Let n be a natural number and b a real number: 1. 𝒃 > 𝟎: 𝐼𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑡ℎ𝑒𝑛 𝑏 ℎ𝑎𝑠 𝑡𝑤𝑜 𝑟𝑒𝑎𝑙 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡𝑠, 𝑒𝑎𝑐ℎ 𝑡ℎ𝑒 𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟; Ex. √𝟗 = ±𝟑 𝟑 𝐼𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑, 𝑡ℎ𝑒𝑛 𝑏 ℎ𝑎𝑠 𝑜𝑛𝑒 𝑟𝑒𝑎𝑙 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡. Ex. √𝟖 = 𝟐 2. 𝐼𝑓 𝒃 = 𝟎: 0 𝑖𝑠 𝑡ℎ𝑒 𝑜𝑛𝑙𝑦 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡 𝑜𝑓 𝑏 = 0. Ex. √𝟎 = 𝟎 𝟑 Ex. √𝟎 = 𝟎 3. 𝐼𝑓 𝒃 < 𝟎: 𝐼𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛, 𝑡ℎ𝑒𝑛 𝑏 ℎ𝑎𝑠 𝑛𝑜 𝑟𝑒𝑎𝑙 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡; Ex. √−𝟗 = 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅 (𝒏𝒐𝒕 𝒂 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓) 𝐼𝑓 𝑛 𝑖𝑠 𝑜𝑑𝑑, 𝑡ℎ𝑒𝑛 𝑏 ℎ𝑎𝑠 𝑜𝑛𝑒 𝑟𝑒𝑎𝑙 𝑛𝑡ℎ 𝑟𝑜𝑜𝑡. 𝟑 Ex. √−𝟖 = −𝟐 Ø Rational Exponents and Radicals: Ex. Evaluate each expression: 𝟏 𝒏 A. 𝒃𝒏 = √𝒃 𝟏 B. 𝟗𝟐 = √𝟗 = 𝟑 6 __________ math114  𝟏 C. √𝟏𝟐𝟏 = 𝟏𝟐𝟏𝟐 = 𝟏𝟏 𝟏 𝟒 D. (−𝟏𝟔)𝟒 = √−𝟏𝟔 = 𝒖𝒏𝒅𝒆𝒇𝒊𝒏𝒆𝒅 (𝒏𝒐𝒕 𝒂 𝒓𝒆𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓) 𝟏 𝟑 E. √−𝟏𝟐𝟓 = (−𝟏𝟐𝟓)𝟑 = −𝟓 𝟏 𝟑 F. (𝟐𝟕)𝟑 = √𝟐𝟕 = 𝟑 𝟏 𝟓 G. √𝟑𝟐 = 𝟑𝟐𝟓 = 𝟐 ___________________________________________________________________________ Ex. Change to radical form: 𝟏 A. (𝟏𝟎𝟎)𝟐 = √𝟏𝟎𝟎 = 𝟏𝟎 𝟏 B. (𝟑𝟐)𝟓 = Ex. Change to rational exponent form: 𝟏 A. √𝟑𝟔𝟏 = (𝟑𝟔𝟏)𝟐 = 𝟏𝟗 𝟏 𝟏 𝟐 𝟐 𝟑 𝟑 B. √𝒙𝟐 + o𝒚𝟐 = (𝒙𝟐 )𝟑 + (𝒚𝟐 )𝟑 = 𝒙𝟑 + 𝒚𝟑 𝟏 𝟑 𝟓 C. 𝟒𝒙 o𝒚𝟑 = 𝟒𝒙(𝒚𝟑 )𝟓 = 𝟒𝒙𝒚𝟓 Ex. Simplify and express answers using positive exponents only? 𝟐 𝟏 𝟐 𝟐 𝟑 A. 𝟖𝟑 = ,𝟖𝟑. = c √𝟖d = 𝟐𝟐 = 𝟐. 𝟐 = 𝟒 𝟒 𝟏 𝟏𝟐.𝟏 𝟏𝟐 B. √𝟑𝟏𝟐 = (𝟑𝟏𝟐 )𝟒 = 𝟑 𝟏.𝟒 = 𝟑 𝟒 = 𝟑𝟑 = 𝟑. 𝟑. 𝟑 = 𝟗. 𝟑 = 𝟐𝟕 7 __________ math114  𝟏 𝟏 𝟏 𝟏 (𝟏)(𝟐)+(𝟑)(𝟏) 𝟐+𝟑 𝟓 C. c𝟑 √𝒙dc𝟐√𝒙d = ,𝟑𝒙𝟑. ,𝟐𝒙𝟐. = (𝟑. 𝟐)𝒙𝟑'𝟐 = 𝟔𝒙 𝟑 (𝟑)(𝟐) = 𝟔𝒙𝟔 = 𝟔𝒙𝟔 𝟏 𝟏 𝟐 𝟏 𝟏𝟏 𝟏 (𝟏)(𝟒)!(𝟔)(𝟏) (!𝟐)(𝟏) 𝟏.. 𝟏 𝟏 𝟒!𝟔 !𝟐 !𝟏 𝟒𝒙𝟑 𝟒 𝟐 𝒙𝟑 𝟐 √𝟒𝒙𝟔 D. p 𝟏 q = 𝟏𝟏 = 𝟏 = 𝟐𝒙𝟔6𝟒 = 𝟐𝒙 (𝟔)(𝟒) = 𝟐𝒙 𝟐𝟒 = 𝟐𝒙 𝟐𝟒 = 𝟐𝒙 (𝟐)(𝟏𝟐) = 𝟐𝒙 𝟏𝟐 =. 𝒙𝟐 𝒙𝟐 𝟐 𝒙𝟒 𝟏 𝟐 𝟐. 𝟏 = 𝟏 𝒙𝟏𝟐 𝒙𝟏𝟐 Ø Simplifying Radicals: Ex. Write in simplified radical form: 𝟏 𝟏 𝟏 𝟏 𝟏 𝟒 𝟖 A. o𝟏𝟔𝒎𝟒 𝒚𝟖 = (𝟏𝟔𝒎𝟒 𝒚𝟖 )𝟐 = 𝟏𝟔𝟏.𝟐 𝒎𝟒.𝟐 𝒚𝟖.𝟐 = 𝟏𝟔𝟐 𝒎𝟐 𝒚𝟐 = √𝟏𝟔𝒎𝟐 𝒚𝟒 = 𝟒𝒎𝟐 𝒚𝟒 𝟏 B. o𝟏𝟐𝒙𝟓 𝒚𝟐 = c𝟏𝟐𝒙𝟓 𝒚𝟐 d𝟐 𝟏 𝟓 𝟐 = 𝟏𝟐𝟐 𝒙𝟐 𝒚𝟐 𝟏 𝟏 = (𝟑. 𝟒)𝟐 𝒙𝟐 𝒙𝟐 𝒚 𝟏 𝟏 = 𝒚𝟑𝟐 𝟒𝟐 𝒙𝟐 √𝒙 = 𝒚√𝟑√𝟒 𝒙𝟐 √𝒙 = 𝟐𝒙𝟐 𝒚√𝟑𝒙 𝟏 𝟔 C. o𝟏𝟔𝒙𝟒 𝒚𝟐 = (𝟏𝟔𝒙𝟒 𝒚𝟐 )𝟔 𝟏 𝟒 𝟐 = 𝟏𝟔𝟔 𝒙𝟔 𝒚𝟔 𝟏 𝟐 𝟏 = (𝟒𝟐 )𝟔 𝒙𝟑 𝒚𝟑 𝟐 𝟏 = 𝟒𝟔 (𝒙𝟐 )𝟑 𝟑o𝒚 𝟏 𝟑 = 𝟒𝟑 √𝒙𝟐 𝟑o𝒚 𝟑 𝟑 𝟑 = √𝟒 √𝒙𝟐 𝟑o𝒚 = o𝟒𝒙𝟐 𝒚 𝟏 𝟓 D. 𝒙 o𝟑𝟔 𝒙𝟕 𝒚𝟏𝟏 = 𝒙 (𝟑𝟔 𝒙𝟕 𝒚𝟏𝟏 )𝟓 𝟏 𝟏 𝟏 = 𝒙 (𝟕𝟐𝟗)𝟓 (𝒙𝟕 )𝟓 (𝒚𝟏𝟏 )𝟓 8 __________ math114  𝟏 𝟕 𝟏𝟏 = 𝒙 (𝟐𝟒𝟑. 𝟑)𝟓 𝒙𝟓 𝒚 𝟓 𝟕 𝟏 𝟏 𝟏 = 𝒙 𝒙𝟓 𝟐𝟒𝟑𝟓 𝟑𝟓 𝒚𝟐 𝒚𝟓 𝟕 = 𝒙𝟏'𝟓 √𝟐𝟒𝟑 √𝟑 𝒚𝟐 𝟓o𝒚 𝟓 𝟓 𝟓+𝟕 =𝒙 𝟓. 𝟑. 𝒚𝟐 𝟓o𝟑𝒚 𝟏𝟐 = 𝟑 𝒙 𝟓 𝒚𝟐 𝟓o𝟑𝒚 𝟐 = 𝟑 𝒙𝟐 𝒙𝟓 𝒚𝟐 𝟓o𝟑𝒚 𝟏 = 𝟑 𝒙𝟐 𝒚𝟐 (𝒙𝟐 )𝟓 𝟓o𝟑𝒚 𝟓 = 𝟑 𝒙𝟐 𝒚𝟐 √𝒙𝟐 𝟓o𝟑𝒚 𝟓 = 𝟑 𝒙𝟐 𝒚𝟐 o𝟑 𝒙𝟐 𝒚 𝟔 𝟔 √𝟐𝒙 𝟔√𝟐𝒙 𝟔√𝟐𝒙 𝟑√𝟐𝒙 E. =. = 𝟐 = = √𝟐𝒙 √𝟐𝒙 √𝟐𝒙 :√𝟐𝒙; 𝟐𝒙 𝒙 𝟏𝟐𝒚𝟐 𝟏𝟐𝒚𝟐 =𝟔𝒚 𝟏𝟐𝒚𝟐 𝟏𝟐𝒚𝟐 (𝟔.𝟐)(𝒚.𝒚) F. =. = 𝟐 = 𝟔𝒚 = 𝟔𝒚 = 𝟐𝒚 =𝟔𝒚 =𝟔𝒚 =𝟔𝒚 :=𝟔𝒚; 𝟏 𝟏 𝟏 𝟒 𝟐 𝟏 𝟏 𝟑 𝟑 𝟑 𝟑 𝟑 𝟖𝒙𝟒 𝟖𝒙𝟒 𝟑 𝟖𝟑 :𝒙𝟒 ;𝟑 √𝟖𝒙𝟑 𝒚𝟑 𝟐𝒙𝒙𝟑 :𝒚𝟐 ;𝟑 𝟐𝒙 √𝒙 =𝒚𝟐 𝟐𝒙 =𝒙𝒚𝟐 G. r =,. = 𝟏 = = = = 𝒚 𝒚 𝒚 𝒚 𝒚 𝒚 𝒚𝟑 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 𝟏 H. −√𝟏𝟐𝟖 = −(𝟏𝟐𝟖)𝟐 = −(𝟏𝟔. 𝟖)𝟐 = −(𝟏𝟔)𝟐 (𝟖)𝟐 = −√𝟏𝟔 (𝟒. 𝟐)𝟐 = −𝟒(𝟒)𝟐 (𝟐)𝟐 = −𝟒√𝟒 √𝟐 = −𝟒. 𝟐. √𝟐 = −𝟖√𝟐 𝟏 𝟏 𝟏 𝟏 I. √𝟐𝟕 − 𝟓√𝟑 = (𝟐𝟕)𝟐 − 𝟓√𝟑 = (𝟗. 𝟑)𝟐 − 𝟓√𝟑 = (𝟗)𝟐 (𝟑)𝟐 − 𝟓√𝟑 = √𝟗√𝟑 − 𝟓√𝟑 = 𝟑√𝟑 − 𝟓√𝟑 = (𝟑 − 𝟓)√𝟑 = −𝟐√𝟑 Ex. Simplify and express answer using positive exponents only: A. 𝒙𝟓 𝒙6𝟐 = B. (𝟐𝒚)(𝟑𝒚𝟐 )(𝟓𝒚𝟒 ) = C. (𝒂𝟐 𝒃𝟑 )𝟓 = 𝟏 𝟓 D. 𝒖𝟑 𝒖𝟑 = 𝟏 𝟏 𝟏 𝟒 6 𝟒(! ) ! 𝒘𝟒 𝟐 𝒘 𝟐 𝟗𝟐 𝒘 𝟐 √𝟗 𝒘!𝟐 𝟑𝒙 E. ,. = 𝟏 𝟏 = 𝟐 = = 𝟗𝒙!𝟐 ! (!𝟐)(! ) ! 𝒙!𝟏 𝒘𝟐 𝟗 𝟐 𝒙 𝟐 𝒙 𝟐 9 __________ math114 R-3 Polynomials: Basic Operations and Factoring Ø Polynomials: Ex. Which of the following are polynomials? A. 𝒙𝟐 − 𝟑𝒙 + 𝟐 … 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 … B. 𝒙𝟒 + √𝟐 … 𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 … 𝟏 C. 𝟐𝒙 + 𝟓 − 𝒙 … 𝑵𝒐𝒏𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 … D. √𝒙𝟑 − 𝟒𝒙 + 𝟏 … 𝑵𝒐𝒏𝑷𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍 … Ex. Given the following polynomials what is the degree of the first term, second term, third term, and the whole polynomial: A. 𝟐𝒙𝟑 − 𝒙𝟔 + 𝟕 The degree of the first term= 𝟑 The degree of the second term= 𝟔 The degree of the third term= 𝟎 The degree of the whole polynomial= 𝟔 B. 𝒙𝟑 𝒚𝟐 + 𝟐𝒙𝟐 𝒚 + 𝟏 The degree of the first term= 𝟓 The degree of the second term= 𝟑 The degree of the third term= 𝟎 The degree of the whole polynomial= 𝟓 Ex. Is the algebraic expression a polynomial? If so, give its degree. A. 𝟒 − 𝒙𝟐 … … … … … … … … … … … B. 𝒙𝟓 − 𝟒𝒙𝟐 + 𝟔6𝟐 … … … … … … … … … ….. C. 𝒙𝟒 + 𝟑𝒙 − √𝟓 … … … … … … … … D. 𝟑𝒙𝟒 − 𝟐𝒙6𝟏 − 𝟏𝟎 … … … … … … … … … … Ø Addition and Subtraction § Adding polynomials: Ex. Add: 𝒙𝟒 − 𝟑𝒙𝟑 + 𝒙𝟐 , −𝒙𝟑 − 𝟐𝒙𝟐 + 𝟑𝒙 , and 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟓 Solution: Vertically: 𝒙𝟒 − 𝟑𝒙𝟑 + 𝒙𝟐 −𝒙𝟑 − 𝟐𝒙𝟐 + 𝟑𝒙 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟓 _______________________ 𝒙𝟒 − 𝟒𝒙𝟑 + 𝟐𝒙𝟐 − 𝒙 − 𝟓 Horizontally: (𝒙𝟒 − 𝟑𝒙𝟑 + 𝒙𝟐 ) + (−𝒙𝟑 − 𝟐𝒙𝟐 + 𝟑𝒙) + (𝟑𝒙𝟐 − 𝟒𝒙 − 𝟓) = 𝒙𝟒 − 𝟑𝒙𝟑 + 𝒙𝟐 − 𝒙𝟑 − 𝟐𝒙𝟐 + 𝟑𝒙 + 𝟑𝒙𝟐 − 𝟒𝒙 − 𝟓 = 𝒙𝟒 + (−𝟑 − 𝟏)𝒙𝟑 + (𝟏 − 𝟐 + 𝟑)𝒙𝟐 + (𝟑 − 𝟒)𝒙 − 𝟓 = 𝒙𝟒 − 𝟒𝒙𝟑 + 𝟐𝒙𝟐 − 𝒙 − 𝟓 10 __________ math114  § Subtracting polynomials: Ex. Subtract: 𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓 𝒇𝒓𝒐𝒎 𝒙𝟐 − 𝟖 Solution: Horizontally: (𝒙𝟐 − 𝟖) − (𝟒𝒙𝟐 − 𝟑𝒙 + 𝟓) = 𝒙𝟐 − 𝟖 − 𝟒𝒙𝟐 + 𝟑𝒙 − 𝟓 = (𝟏 − 𝟒)𝒙𝟐 + 𝟑𝒙 + (−𝟖 − 𝟓) = −𝟑𝒙𝟐 + 𝟑𝒙 − 𝟏𝟑 Ø Multiplication § Multiplying polynomials: Ex. Multiply: A. (𝟐𝒙 − 𝟑)(𝟑𝒙𝟐 − 𝟐𝒙 + 𝟑) Solution: = 𝟐𝒙(𝟑𝒙𝟐 − 𝟐𝒙 + 𝟑) − 𝟑(𝟑𝒙𝟐 − 𝟐𝒙 + 𝟑) = 𝟐𝒙(𝟑𝒙𝟐 ) + 𝟐𝒙(−𝟐𝒙) + 𝟐𝒙(𝟑) − 𝟑(𝟑𝒙𝟐 ) − 𝟑(−𝟐𝒙) − 𝟑(𝟑) = 𝟔𝒙𝟑 − 𝟒𝒙𝟐 + 𝟔𝒙 − 𝟗𝒙𝟐 + 𝟔𝒙 − 𝟗 = 𝟔𝒙𝟑 + (−𝟒 − 𝟗)𝒙𝟐 + (𝟔 + 𝟔)𝒙 − 𝟗 = 𝟔𝒙𝟑 − 𝟏𝟑𝒙𝟐 + 𝟏𝟐𝒙 − 𝟗 B. (𝒂 − 𝒃)(𝒂𝟐 + 𝒂𝒃 + 𝒃𝟐 ) = … … … … … … … Ex. Perform the indicated operations and simplify C. (𝟒𝒙 − 𝒚)𝟐 = (𝟒𝒙 − 𝒚)(𝟒𝒙 − 𝒚) 𝑶𝒓 (𝟒𝒙 − 𝒚)𝟐 = (𝟒𝒙)𝟐 − 𝟐(𝟒𝒙)(𝒚) + 𝒚𝟐 = 𝟒𝒙(𝟒𝒙 − 𝒚) − 𝒚(𝟒𝒙 − 𝒚) = 𝟒𝟐 𝒙𝟐 − 𝟖𝒙𝒚 + 𝒚𝟐 = 𝟒𝒙(𝟒𝒙) + 𝟒𝒙(−𝒚) − 𝒚(𝟒𝒙) − 𝒚(−𝒚) = 𝟏𝟔𝒙𝟐 − 𝟖𝒙𝒚 + 𝒚𝟐 𝟐 𝟐 = 𝟏𝟔𝒙 − 𝟒𝒙𝒚 − 𝟒𝒙𝒚 + 𝒚 = 𝟏𝟔𝒙𝟐 − 𝟖𝒙𝒚 + 𝒚𝟐 D. (𝟓𝒚 − 𝟏)(𝟑 − 𝟐𝒚) = … … … … … … … … ….. E. 𝟐𝒚 − 𝟑𝒚[𝟒 − 𝟐(𝒚 − 𝟏)] = 𝟐𝒚 − 𝟑𝒚[𝟒 − 𝟐𝒚 + 𝟐] = 𝟐𝒚 − 𝟑𝒚[𝟔 − 𝟐𝒚] = 𝟐𝒚 − 𝟑𝒚(𝟔) − 𝟑𝒚(−𝟐𝒚) = 𝟐𝒚 − 𝟏𝟖𝒚 + 𝟔𝒚𝟐 = −𝟏𝟔𝒚 + 𝟔𝒚𝟐 F. 𝟐(𝒙 − 𝟏) + 𝟑(𝟐𝒙 − 𝟑) − (𝟒𝒙 − 𝟓) = … … … … … … … …. Ex. Remove Parenthesis and combine like terms: A. 𝟐(𝟑𝒙𝟐 − 𝟐𝒙 + 𝟓) + (𝒙𝟐 + 𝟑𝒙 − 𝟕) = 𝟐(𝟑𝒙𝟐 ) + 𝟐(−𝟐𝒙) + 𝟐(𝟓) + 𝒙𝟐 + 𝟑𝒙 − 𝟕 = 𝟔𝒙𝟐 − 𝟒𝒙 + 𝟏𝟎 + 𝒙𝟐 + 𝟑𝒙 − 𝟕 = (𝟔 + 𝟏)𝒙𝟐 + (−𝟒 + 𝟑)𝒙 + (𝟏𝟎 − 𝟕) = 𝟕𝒙𝟐 − 𝒙 + 𝟑 Ø Factoring: Ex. 𝟏𝟑 = 𝟏𝟑. 𝟏 (𝒑𝒓𝒊𝒎𝒆 𝒏𝒖𝒎𝒃𝒆𝒓) 𝟑𝟎 = 𝟔. 𝟓 = 𝟐. 𝟑. 𝟓 (𝒄𝒐𝒎𝒑𝒐𝒔𝒊𝒕𝒆 𝒏𝒖𝒎𝒃𝒆𝒓) 𝒙𝟐 − 𝟒 = (𝒙 − 𝟐)(𝒙 + 𝟐) 𝑵𝒐𝒕 𝒑𝒓𝒊𝒎𝒆 𝒙 − 𝟐 = 𝑷𝒓𝒊𝒎𝒆 𝒃𝒆𝒄𝒂𝒖𝒔𝒆 𝒊𝒕 𝒄𝒂𝒏@ 𝒕 𝒃𝒆 𝒘𝒓𝒊𝒕𝒕𝒆𝒏 𝒂𝒔 𝒂 𝒑𝒓𝒐𝒅𝒖𝒄𝒕 𝒐𝒇 𝒕𝒘𝒐 𝒑𝒐𝒍𝒚𝒏𝒐𝒎𝒊𝒂𝒍𝒔 𝟐 11 __________ math114  § Factoring out common factors: Ex. Factor out, relative to the integers, all factors common to all terms: A. 𝟐𝒙𝟑 𝒚 − 𝟖𝒙𝟐 𝒚𝟐 − 𝟔𝒙𝒚𝟑 = 𝟐𝒙𝒚(𝒙𝟐 − 𝟒𝒙𝒚 − 𝟑𝒚𝟐 ) B. 𝟔𝒙𝟒 − 𝟖𝒙𝟑 − 𝟐𝒙𝟐 = … … … … … … … … … … … …. C. 𝒙𝟐 𝒚 + 𝟐𝒙𝒚𝟐 + 𝒙𝟐 𝒚𝟐 = … … … … … … … … … … …. D. 𝟐𝒙(𝟑𝒙 − 𝟐) − 𝟕(𝟑𝒙 − 𝟐) = (𝟑𝒙 − 𝟐)(𝟐𝒙 − 𝟕) E. 𝟐𝒘(𝒚 − 𝟐𝒛) − 𝒙(𝒚 − 𝟐𝒛) = … … … … … … … § Factoring by grouping: Ex. Factor completely, relative to the integers, by grouping: A. 𝟑𝒙𝟐 − 𝟔𝒙 + 𝟒𝒙 − 𝟖 = (𝟑𝒙𝟐 − 𝟔𝒙) + (𝟒𝒙 − 𝟖) = 𝟑𝒙(𝒙 − 𝟐) + 𝟒(𝒙 − 𝟐) = (𝒙 − 𝟐)(𝟑𝒙 + 𝟒) B. 𝒘𝒚 + 𝒘𝒛 − 𝟐𝒙𝒚 − 𝟐𝒙𝒛 = … … … … … … … …. C. 𝒙𝟐 + 𝟒𝒙 + 𝒙 + 𝟒 = (𝒙𝟐 + 𝒙) + (𝟒𝒙 + 𝟒) = 𝒙(𝒙 + 𝟏) + 𝟒(𝒙 + 𝟏) = (𝒙 + 𝟏)(𝒙 + 𝟒) D. 𝟑𝒂𝟐 − 𝟏𝟐𝒂𝒃 − 𝟐𝒂𝒃 + 𝟖𝒃𝟐 = (𝟑𝒂𝟐 − 𝟏𝟐𝒂𝒃) + (−𝟐𝒂𝒃 + 𝟖𝒃𝟐 ) = 𝟑𝒂(𝒂 − 𝟒𝒃) + 𝟐𝒃(−𝒂 + 𝟒𝒃) = 𝟑𝒂(𝒂 − 𝟒𝒃) − 𝟐𝒃(𝒂 − 𝟒𝒃) = (𝒂 − 𝟒𝒃)(𝟑𝒂 − 𝟐𝒃) E. 𝟖𝒂𝒄 + 𝟑𝒃𝒅 − 𝟔𝒃𝒄 − 𝟒𝒂𝒅 = … … … … … … … … … … … … … … …. Ex. Factor completely, relative to the integers: A. 𝒙𝟐 − 𝒙𝒚 + 𝟑𝒙𝒚 − 𝟑𝒚𝟐 = (𝒙𝟐 − 𝒙𝒚) + (𝟑𝒙𝒚 − 𝟑𝒚𝟐 ) = 𝒙(𝒙 − 𝒚) + 𝟑𝒚(𝒙 − 𝒚) = (𝒙 − 𝒚)(𝒙 + 𝟑𝒚) § Factoring second (2ed) degree polynomials: Ex. Factor each polynomial, if possible, using integer coefficients: A. 2𝑥 1 + 3𝑥𝑦 − 2𝑦 1 = (𝟐𝒙 − 𝒚)(𝒙 + 𝟐𝒚) 𝑪𝒉𝒆𝒄𝒌: (𝟐𝒙 − 𝒚)(𝒙 + 𝟐𝒚) = 𝟐𝒙(𝒙 + 𝟐𝒚) − 𝒚(𝒙 + 𝟐𝒚) = 𝟐𝒙𝟐 + 𝟒𝒙𝒚 − 𝒙𝒚 − 𝟐𝒚𝟐 = 2𝑥 1 + 3𝑥𝑦 − 2𝑦 1 B. 6𝑥 1 + 5𝑥𝑦 − 4𝑦 1 = (𝟑𝒙 + 𝟒𝒚)(𝟐𝒙 − 𝟏𝒚) 𝑪𝒉𝒆𝒄𝒌: (𝟑𝒙 + 𝟒𝒚)(𝟐𝒙 − 𝒚) = 𝟑𝒙(𝟐𝒙 − 𝒚) + 𝟒𝒚(𝟐𝒙 − 𝒚) = 𝟔𝒙𝟐 − 𝟑𝒙𝒚 + 𝟖𝒙𝒚 − 𝟒𝒚𝟐 = 6𝑥 1 + 5𝑥𝑦 − 4𝑦 1 C. 𝑥 1 − 3𝑥 + 4 = (𝒙 − 𝟐)(𝒙 − 𝟐) 𝑪𝒉𝒆𝒄𝒌: (𝒙 − 𝟐)(𝒙 − 𝟐) = 𝒙(𝒙 − 𝟐) − 𝟐(𝒙 − 𝟐) = 𝒙𝟐 − 𝟐𝒙 − 𝟐𝒙 + 𝟒 12 __________ math114  = 𝑥 1 − 4𝑥 + 4 × 𝑥 1 − 3𝑥 + 4 = (𝒙 − 𝟒)(𝒙 − 𝟏) 𝑪𝒉𝒆𝒄𝒌: (𝒙 − 𝟒)(𝒙 − 𝟏) = 𝒙(𝒙 − 𝟏) − 𝟒(𝒙 − 𝟏) = 𝒙𝟐 − 𝒙 − 𝟒𝒙 + 𝟒 = 𝑥 1 − 5𝑥 + 4 × ∴ 𝑥 1 − 3𝑥 + 4 𝑖𝑠 𝑛𝑜𝑡 𝑓𝑎𝑐𝑡𝑜𝑟𝑎𝑏𝑙𝑒. Ex. Factor completely, relative to the integers. If a polynomial is prime relative to the integers, say so. A. 2𝑥 1 + 𝑥 − 3 = (𝟐𝒙 + 𝟑)(𝒙 − 𝟏) 𝑪𝒉𝒆𝒄𝒌: (𝟐𝒙 + 𝟑)(𝒙 − 𝟏) = 𝟐𝒙(𝒙 − 𝟏) + 𝟑(𝒙 − 𝟏) = 𝟐𝒙𝟐 − 𝟐𝒙 + 𝟑𝒙 − 𝟑 = 2𝑥 1 + 𝑥 − 3 B. 𝒙𝟐 + 𝟓𝒙𝒚 − 𝟏𝟒𝒚𝟐 = … … … … … … … … …. C. 𝟔𝒎𝟐 − 𝒎𝒏 − 𝟏𝟐𝒏𝟐 = … … … … … … … …. § Factoring by using special factoring formulas: Ex. Factor completely relative to the integers: A. 𝒙𝟐 + 𝟔𝒙𝒚 + 𝟗𝒚𝟐 = 𝒙𝟐 + 𝟐(𝒙)(𝟑𝒚) + 𝟗𝒚𝟐 = (𝒙 + 𝟑𝒚)𝟐 B. 𝒙𝟐 − 𝟔𝒙𝒚 + 𝟗𝒚𝟐 = 𝒙𝟐 − 𝟐(𝒙)(𝟑𝒚) + 𝟗𝒚𝟐 = (𝒙 − 𝟑𝒚)𝟐 C. 𝟗𝒙𝟐 − 𝟒𝒚𝟐 = (𝟑𝒙 − 𝟐𝒚)(𝟑𝒙 − 𝟐𝒚) D. 𝒙𝟑 − 𝒚𝟑 = (𝒙 − 𝒚)(𝒙𝟐 + 𝒙𝒚 + 𝒚𝟐 ) E. 𝒎𝟑 + 𝒏𝟑 = (𝒎 + 𝒏)(𝒎𝟐 − 𝒎𝒏 + 𝒏𝟐 ) Ex. Factor completely, relative to the integers. If a polynomial is prime relative to the integers, say so. A. 𝟏𝟔𝒙𝟐 − 𝟐𝟓 = … … … … … …. B. 𝟖𝒎𝟑 − 𝟏 = 𝟐𝟑 𝒎𝟑 − 𝟏 = (𝟐𝒎 − 𝟏)((𝟐𝒎)𝟐 + 𝟐𝒎(𝟏) + 𝟏𝟐 ) = (𝟐𝒎 − 𝟏)(𝟐𝟐 𝒎𝟐 + 𝟐𝒎 + 𝟏)) = (𝟐𝒎 − 𝟏)(𝟒𝒎𝟐 + 𝟐𝒎 + 𝟏) C. 𝒙 + 𝒚 𝒛 = (𝒙 + 𝒚)(𝒙𝟐 − 𝒙𝒚𝒛 + 𝒚𝟐 𝒛𝟐 ) 𝟑 𝟑 𝟑 13 __________ math114 Chapter 1: Equations and Inequalities 1-1 Linear Equations and Applications Ø Solving Linear Equations: Ex. Solve each equation: A. 𝟓𝒙 − 𝟗 = 𝟑𝒙 + 𝟕 𝟓𝒙 − 𝟑𝒙 = 𝟕 + 𝟗 𝟐𝒙 = 𝟏𝟔 𝟐𝒙 𝟏𝟔 = 𝟐 𝟐 𝒙=𝟖 𝑪𝒉𝒆𝒄𝒌: 𝟓𝒙 − 𝟗 = 𝟑𝒙 + 𝟕 (𝒙 = 𝟖) 𝟓(𝟖) − 𝟗 = 𝟑(𝟖) + 𝟕 𝟒𝟎 − 𝟗 = 𝟐𝟒 + 𝟕 𝟑𝟏 = 𝟑𝟏 Ex. Solve each equation: A. 𝟏𝟎𝒙 − 𝟕 = 𝟒𝒙 − 𝟐𝟓 Check: 𝟏𝟎𝒙 − 𝟕 = 𝟒𝒙 − 𝟐𝟓(𝒙 = −𝟑) 𝟏𝟎𝒙 − 𝟒𝒙 = −𝟐𝟓 + 𝟕 𝟏𝟎(−𝟑) − 𝟕 = 𝟒(−𝟑) − 𝟐𝟓 𝟔𝒙 = −𝟏𝟖 −𝟑𝟎 − 𝟕 = −𝟏𝟐 − 𝟐𝟓 𝟔𝒙 $𝟏𝟖 𝟔𝒙 = 𝟔 −𝟑𝟕 = −𝟑𝟕 𝒙 = −𝟑 B. 𝟑(𝒙 + 𝟐) = 𝟓(𝒙 − 𝟔) 𝟑(𝒙) + 𝟑(𝟐) = 𝟓(𝒙) + 𝟓(−𝟔) 𝟑𝒙 + 𝟔 = 𝟓𝒙 − 𝟑𝟎 𝟑𝒙 − 𝟓𝒙 = −𝟑𝟎 − 𝟔 −𝟐𝒙 = −𝟑𝟔 6𝟐𝒙 6𝟑𝟔 = 6𝟐 6𝟐 𝒙 = +𝟏𝟖 Check: 𝟑(𝒙 + 𝟐) = 𝟓(𝒙 − 𝟔) (𝒙 = 𝟏𝟖) 𝟑(𝟏𝟖 + 𝟐) = 𝟓(𝟏𝟖 − 𝟔) 𝟑(𝟐𝟎) = 𝟓(𝟏𝟐) 𝟔𝟎 = 𝟔𝟎 14 __________ math114  𝟑𝒂6𝟒 𝟕6𝟐𝒂 C. 𝟓 − = 𝟓 𝟐 𝟑𝒂6𝟒 𝟕6𝟐𝒂 (𝟏𝟎)(𝟓) − (𝟏𝟎) ,. = (𝟏𝟎) ,. 𝟓 𝟐 𝟓𝟎 − 𝟐(𝟑𝒂 − 𝟒) = 𝟓(𝟕 − 𝟐𝒂) 𝟓𝟎 − 𝟔𝒂 + 𝟖 = 𝟑𝟓 − 𝟏𝟎𝒂 𝟓𝟖 − 𝟔𝒂 = 𝟑𝟓 − 𝟏𝟎𝒂 𝟏𝟎𝒂 − 𝟔𝒂 = 𝟑𝟓 − 𝟓𝟖 𝟒𝒂 = −𝟐𝟑 6𝟐𝟑 𝒂= 𝟒 𝟑𝒂$𝟒 𝟕$𝟐𝒂 $𝟐𝟑 Check: 𝟓 − 𝟓 = 𝟐 (𝒂 = 𝟒 ) 𝟏 𝟏 𝟓 − (𝟑𝒂 − 𝟒) = (𝟕 − 𝟐𝒂) 𝟓 𝟐 𝟏 $𝟐𝟑 𝟏 $𝟐𝟑 𝟓 − 7𝟑 7 8 − 𝟒8 = 𝟐 (𝟕 − 𝟐 7 8) 𝟓 𝟒 𝟒 𝟏 $𝟔𝟗 𝟏 𝟒𝟔 𝟓−𝟓7 𝟒 − 𝟒8 = (𝟕 + 𝟐 𝟒 ) 𝟏 $𝟔𝟗(𝟏)$𝟒(𝟒) 𝟏 𝟕(𝟒)3𝟒𝟔(𝟏) 𝟓−𝟓7 𝟒(𝟏) 8= ( 𝟐 𝟏(𝟒) ) 𝟏 $𝟔𝟗$𝟏𝟔 𝟏 𝟐𝟖3𝟒𝟔 𝟓−𝟓7 𝟒 8 = 𝟐( 𝟒 ) 𝟏 $𝟖𝟓 𝟏 𝟕𝟒 𝟓− 7 8= ( ) 𝟓 𝟒 𝟐 𝟒 𝟖𝟓 𝟕𝟒 𝟓 + 𝟐𝟎 = 𝟖 𝟓(𝟐𝟎)3𝟏(𝟖𝟓) 𝟕𝟒 = 𝟏(𝟐𝟎) 𝟖 𝟏𝟎𝟎3𝟖𝟓 𝟕𝟒 = 𝟐𝟎 𝟖 𝟏𝟖𝟓 𝟕𝟒 = 𝟐𝟎 𝟖 𝟓.𝟑𝟕 𝟐.𝟑𝟕 = 𝟓.𝟒 𝟐.𝟒 𝟑𝟕 𝟑𝟕 𝟒 = 𝟒 1-2 Linear Inequalities Ø Understanding Inequality and Interval Notation: Ex: Rewrite each of the following in inequality notation and graph on a real number line: A. [−𝟐, 𝟑) Solution: −𝟐 ≤ 𝒙 < 𝟑 B. (−𝟒, 𝟐) Solution: −𝟒 < 𝒙 < 𝟐 C. [−𝟐, ∞) Solution: 𝒙 ≥ −𝟐 D. (−∞, 𝟑) Solution: 𝒙 < 𝟑 E. [−𝟖, 𝟕] Solution: ………………. F. [−𝟔, ∞) Solution: ………………. 15 __________ math114 Ex. Rewrite each of the following in interval notation and graph on a real number line: A. −𝟐 < 𝒙 ≤ 𝟔 Solution: (−𝟐, 𝟔] The graph: …………………………….. B. 𝒙 ≤ −𝟐 Solution: (−∞, −𝟐] The graph: ……………………………. Ex. Rewrite each of the following in interval notation and in inequality notation: A. Solution: in interval notation [−𝟕, 𝟐) in inequality notation −𝟕 ≤ 𝒙 < 𝟐 Ex. If 𝑨 = (−𝟐, 𝟓) and 𝑩 = (𝟏, ∞), graph the sets𝑨 ∪ 𝑩 𝒂𝒏𝒅 𝑨 ∩ 𝑩 and write them in interval notation: Solution: Ø Solving Linear Inequalities: Ex. Solve and graph: A. 𝟕𝒙 − 𝟖 < 𝟒𝒙 + 𝟕 𝟕𝒙 − 𝟒𝒙 < 𝟕 + 𝟖 𝟑𝒙 < 𝟏𝟓 𝟑𝒙 𝟏𝟓 < 𝟑 𝟑 𝒙 < 𝟓 , 𝒙 ∈ (−∞, 𝟓) The graph: ……………………………. B. 𝟐(𝟐𝒙 + 𝟑) − 𝟏𝟎 < 𝟔(𝒙 − 𝟐) 𝟒𝒙 + 𝟔 − 𝟏𝟎 < 𝟔𝒙 − 𝟏𝟐 𝟒𝒙 − 𝟒 < 𝟔𝒙 − 𝟏𝟐 𝟒𝒙 − 𝟔𝒙 < −𝟏𝟐 + 𝟒 −𝟐𝒙 < −𝟖 6𝟐𝒙 6𝟖 6𝟐 > 6𝟐 𝒙 > 𝟒 , 𝒙 ∈ (𝟒, ∞) 16 __________ math114 The graph: 𝟐𝒙6𝟑 𝟒𝒙 C. +𝟔≥𝟐+ 𝟒 𝟑 𝟐𝒙6𝟑 𝟒𝒙 (𝟏𝟐) ,. + 𝟏𝟐(𝟔) ≥ 𝟏𝟐(𝟐) + (𝟏𝟐) ,. 𝟒 𝟑 𝟑(𝟐𝒙 − 𝟑) + 𝟕𝟐 ≥ 𝟐𝟒 + 𝟒(𝟒𝒙) 𝟔𝒙 − 𝟗 + 𝟕𝟐 ≥ 𝟐𝟒 + 𝟏𝟔𝒙 𝟔𝒙 + 𝟔𝟑 ≥ 𝟐𝟒 + 𝟏𝟔𝒙 𝟔𝒙 − 𝟏𝟔𝒙 ≥ 𝟐𝟒 − 𝟔𝟑 −𝟏𝟎𝒙 ≥ 𝟑𝟗 6𝟏𝟎𝒙 𝟑𝟗 ≤ 6𝟏𝟎 6𝟏𝟎 𝟑𝟗 𝟑𝟗 𝒙 ≤ 6𝟏𝟎 , 𝒙 ∈ (−∞, −𝟑. 𝟗) (6𝟏𝟎 = −𝟑. 𝟗) D. −𝟑 ≤ 𝟒 − 𝟕𝒙 < 𝟏 −𝟑 − 𝟒 ≤ 𝟒 − 𝟒 − 𝟕𝒙 < 𝟏𝟖 − 𝟒 −𝟕 ≤ −𝟕𝒙 < 𝟏𝟒 6𝟕 6𝟕𝒙 𝟏𝟒 ≥ > 6𝟕 6𝟕 6𝟕 𝟏 ≥ 𝒙 > −𝟐 , 𝒙(−𝟐, 𝟏] E. −𝟒 < 𝟓𝒕 + 𝟔 ≤ 𝟐𝟏 Solution: ……………………….. 1-5 Quadratic Equations (𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎, 𝒂 ≠ 𝟎) Ø Solving quadratic equations by factoring: Ex: Solve each equation by factoring: A. (𝒙 − 𝟓)(𝒙 + 𝟑) = 𝟎 Solution: 𝑥 − 5 = 0 𝑂𝑟 𝑥 + 3 = 0 𝑥 = +5 𝑜𝑟 𝑥 = −3 Solution set = {−3, +5} B. 𝟔𝒙𝟐 − 𝟏𝟗𝒙 − 𝟕 = 𝟎 (2𝑥 − 7)(3𝑥 + 1) = 0 Check: 6𝑥 ! − 19𝑥 − 7 = (2𝑥 − 7)(3𝑥 + 1) 2𝑥 − 7 = 0 𝑂𝑟 3𝑥 + 1 = 0 = 2𝑥(3𝑥 + 1) − 7(3𝑥 + 1) 2𝑥 = +7 𝑜𝑟 3𝑥 = −1 = 6𝑥 ! + 2𝑥 − 21𝑥 − 7 45 6 75 $8 = 𝑂𝑟 = = 6𝑥 ! − 19𝑥 − 7 4 4 7 7 6 $8 𝑥 = 4 𝑂𝑟 𝑥 = 7 $8 6 Solution set= { 7 , 4} 17 __________ math114  C. 𝒙𝟐 − 𝟔𝒙 + 𝟓 = −𝟒 𝑥 4 − 6𝑥 + 5 + 4 = 0 𝑥 4 − 6𝑥 + 9 = 0 (𝑥 − 3)(𝑥 − 3) = 0 Check: 𝑥 ! − 6𝑥 + 9 = (𝑥 − 3)(𝑥 − 3) 𝑥 − 3 = 0 𝑂𝑟 𝑥 − 3 = 0 = 𝑥(𝑥 − 3) − 3(𝑥 − 3) 𝑥 = +3 𝑂𝑟 𝑥 = +3 = 𝑥 ! − 3𝑥 − 3𝑥 + 9 Solution set= {3} = 𝑥 ! − 6𝑥 + 9 D. 𝟑𝒘𝟐 + 𝟏𝟑𝒘 = 𝟏𝟎 (Practice: the students can solve it by the same idea of C) E. 𝟐𝒙𝟐 = 𝟑𝒙 2𝑥 4 − 3𝑥 = 0 𝑥(2𝑥 − 3) = 0 𝑥 = 0 𝑂𝑟 2𝑥 − 3 = 0 𝑥 = 0 0𝑟 2𝑥 = +3 45 7 𝑥 = 0 𝑂𝑟 4 = 4 3 𝑥 = 0 𝑂𝑟 𝑥 = 2 7 Solution set= {0, } 4 F. 𝟐𝒙𝟐 = 𝟖𝒙 (Practice: the students can solve it by the same idea of E) G. −𝟖 = 𝟐𝟐𝒕 − 𝟔𝒕𝟐 6𝑡 4 − 22𝑡 − 8 = 0 (2𝑡 − 8)(3𝑡 + 1) = 0 Check: 6𝑡 ! − 22𝑡 − 8 = (2𝑡 − 8)(3𝑡 + 1) 2𝑡 = +8 𝑂𝑟 3𝑡 = −1 = 2𝑡(3𝑡 + 1) − 8(3𝑡 + 1) 49 : 79 $8 4 = 4 𝑂𝑟 7 = 7 = 6𝑡 ! + 2𝑡 − 24𝑡 − 8 $8 𝑡 = 4 𝑂𝑟 𝑡 = = 6𝑡 ! − 22𝑡 − 8 7 $8 Solution set= { , 4} 7 Ø Solving quadratic equations by quadratic formula: 6𝒃±=𝒃𝟐 6𝟒𝒂𝒄 If 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎 then 𝒙 = 𝟐𝒂 Ex. Solve each equation using the quadratic formula: A. 𝒙𝟐 = 𝟑𝒙 + 𝟏 𝑥 " − 3𝑥 − 1 = 0 𝑎 = 1, 𝑏 = −3, 𝑐 = −1 #$±√$ ! #'() 𝑥= "( −(−3) ± 3(−3)" − 4(1)(−1) 𝑥= 2(1) +3 ± √9 + 4 𝑥= 2 3 ± √13 𝑥= 2 7$√87 73√87 Solution set= { , } 4 4 18 __________ math114  B. 𝒙𝟐 − 𝟐𝒙 − 𝟏 = 𝟎 𝑎 = 1, 𝑏 = −2, 𝑐 = −1 −𝑏 ± √𝑏/ − 4𝑎𝑐 𝑥= 2𝑎 −(−2) ± 1(−2)/ − 4(1)(−1) 𝑥= 2(1) 2 ± √4 + 4 𝑥= 2 2 ± √8 𝑥= 2 2 ± √4.2 𝑥= 2 2 ± √4√2 𝑥= 2 2 ± 2√2 𝑥= 2 2(1 ± √2 ) 𝑥= 2 𝑥 = 1 ± √2 Solution set= {1 − √2 , 1 + √2} 𝟑 C. 𝟐𝒙 + = 𝒙𝟐 𝟐 3 (2)(2𝑥) + (2) 8 9 = (2)(𝑥 " ) 2 4𝑥 + 3 = 2𝑥 " 2𝑥 4 − 4𝑥 − 3 = 0 𝑎 = 2, 𝑏 = −4, 𝑐 = −3 01±31 ! 0456 𝑥= /5 −(−4) ± 1(−4)/ − 4(2)(−3) 𝑥= 2(2) +4 ± √16 + 24 𝑥= 4 4 ± √40 𝑥= 4 4 ± √4.10 𝑥= 4 4 ± √4√10 𝑥= 4 4 ± 2√10 𝑥= 4 2(2 ± √10 ) 𝑥= 4 2 ± √10 𝑥= 2 4$√8< 43√8< Solution set= { 4 , 4 } 19 __________ math114 Ex. Solve each equation by any method: A. 𝟏𝟐𝒙𝟐 + 𝟕𝒙 = 𝟏𝟎 4 12𝑥 + 7𝑥 − 10 = 0 (3𝑥 − 2)(4𝑥 + 5) = 0 Check: 12𝑥 ! + 7𝑥 − 10 = (3𝑥 − 2)(4𝑥 + 5) 3𝑥 − 2 = 0 𝑂𝑟 4𝑥 + 5 = 0 = 3𝑥(4𝑥 + 5) − 2(4𝑥 + 5) 3𝑥 = +2 𝑂𝑟 4𝑥 = −5 = 12𝑥 ! + 15𝑥 − 8𝑥 − 10 *+ " '+ #, * = * 𝑂𝑟 ' = ' = 12𝑥 ! + 7𝑥 − 10 2 −5 𝑥 = 𝑂𝑟 𝑥 = 3 4 #, " Solution set= { ' , *} B. (𝟐𝒚 − 𝟑)𝟐 = 𝟓 24 𝑦 4 − 2(2𝑦)(3) + 34 = 5 4𝑦 4 − 12𝑦 + 9 = 5 4𝑦 4 − 12𝑦 + 9 − 5 = 0 4𝑦 4 − 12𝑦 + 4 = 0 𝑎 = 4, 𝑏 = −12, 𝑐 = 4 $=±√= : $?@A 𝑦= 4@ $($84)±B($84): $?(?)(?) 𝑦= 4(?) 384±√8??$C? 𝑦= : 84±√8??$C? 𝑦= : 84±√:< 𝑦= : 84±√8C.D 𝑦= : 84±√8C√D 𝑦= : 84±?√D 𝑦= : 4(7±√9) 𝑦= : 7±√D 𝑦= 4 7$√D 73√D Solution set= { 4 , 4 } 20 __________ math114  Chapter 3: Functions 3-1 Functions Ø Definition of a function: Ex. Determine whether each set specifies a function. If it does, then state the domain and the range? A. S= {(1,4), (2,3), (3,2), (4,3), (5,4)} Solution: yes, it is a function, because all the ordered pairs in S have distinct first components. 𝐷𝑜𝑚𝑎𝑖𝑛 = {1, 2, 3, 4, 5} 𝑅𝑎𝑛𝑔𝑒 = {4,3,2} B. T= {(1,4), (2,3), (3,2), (2,4), (1,5)} Solution: No, it is not a function, because there are ordered pairs in T with the same first component [for example, (1, 4) and (1, 5)]. C. {(10, -10), (5, -5), (0,0), (5,5), (10,10)} (practice: the students can solve C by the same idea of B) Ex. Indicate whether each set defines a function. Find the domain and the range of each function? A. {(-1,4), (0,3), (1,2), (2,1)} Solution: yes, it is a function, because all the ordered pairs in this set have distinct first components. 𝐷𝑜𝑚𝑎𝑖𝑛 = {−1, 0, 1, 2} 𝑅𝑎𝑛𝑔𝑒 = {4, 3, 2, 1} B. {(2,4), (3,6), (4,8), (5,10)} (practice: the students can solve B by the same idea of A) Ø Definition a function by equations: Ex: 𝐹(𝑥) = 𝑥 4 − 4 𝑦 = 𝑥4 − 4 Evaluating Function: 𝟏𝟓 Ex.1: A. Find 𝑓(6), 𝑓(𝑎), 𝑎𝑛𝑑 𝑓(6 + 𝑎) 𝑓𝑜𝑟 𝒇(𝒙) = 𝒙$𝟑 8D 8D 𝒇(𝟔) = 𝟔$7 = 7 =5 15 𝒇(𝒂) = 𝒂−3 15 15 𝒇(𝟔 + 𝒂) = = 𝟔+𝒂−3 𝑎+3 B. Find 𝑔(7), 𝑔(ℎ), 𝑎𝑛𝑑 𝑔(7 + ℎ) 𝑓𝑜𝑟 𝒈(𝒙) = 𝟏𝟔 + 𝟑𝒙 − 𝒙𝟐 𝒈(𝟕) = 16 + 3(𝟕) − (𝟕)/ = 16 + 21 − 49 = 37 − 49 = −12 𝒈(𝒉) = 16 + 3(𝒉) − 𝒉/ = 16 + 3ℎ − ℎ/ 𝒈(𝟕 + 𝒉) = 16 + 3(𝟕 + 𝒉) − (𝟕 + 𝒉)/ = 16 + 21 + 3ℎ − (7/ + 2(7)(ℎ) + ℎ/ ) = 16 + 21 + 3ℎ − (49 + 14ℎ + ℎ/ ) = 16 + 21 + 3ℎ − 49 − 14ℎ − ℎ/ = −12 − 11ℎ − ℎ/ 21 __________ math114  𝟐 C. Find 𝐾(9), 4𝐾(𝑎), 𝑎𝑛𝑑 𝐾(4𝑎) 𝑓𝑜𝑟 𝑲(𝒙) = √𝒙0𝟐 2 2 2 𝑲(𝟗) = = = =2 √9 − 2 3 − 2 1 2 4∙2 8 𝟒𝑲(𝒂) = 4 O P= = √𝑎 − 2 √𝑎 − 2 √𝑎 − 2 2 2 2 2 1 𝑲(𝟒𝒂) = = = = = √4𝑎 − 2 √4√𝑎 − 2 2√𝑎 − 2 2(√𝑎 − 1) √𝑎 − 1 Ex.2: A. Let 𝒇(𝒙) = 𝟑𝒙 − 𝟓 𝑓𝑖𝑛𝑑 𝑓(3), 𝑓(ℎ), 𝑓(3) + 𝑓(ℎ), 𝑓(3 + ℎ) 𝒇(𝟑) = 3(𝟑) − 5 = 9 − 5 = 4 𝒇(𝒉) = 3𝒉 − 5 𝒇(𝟑) + 𝒇(𝒉) = 4 + (3ℎ − 5) = 4 + 3ℎ − 5 = −1 + 3ℎ 𝒇(𝟑 + 𝒉) = 3(𝟑 + 𝒉) − 5 = 9 + 3ℎ − 5 = 4 + 3ℎ B. Let 𝑓(𝑤) = −𝑤 / + 2𝑤 𝑓𝑖𝑛𝑑 𝑓(4), 𝑓(−4), 𝑓(4 + 𝑎), 𝑓(2 − 𝑎) (practice: the students can solve B by the same idea of Ex.1 & Ex.2) Finding the Domain of a Function: Ex: Find the domain of the function defined by the equation: 𝑦 = √𝑥 − 3 , assuming x is the independent variable. Solution: for 𝑦 to be real, 𝑥 − 3 ≥ 0 ⇒ 𝑥 ≥ +3 ∴ The domain of y is 𝑥 ≥ 3 𝑜𝑟 𝑥 ∈ [3, ∞) 𝑜𝑟 {𝑥|𝑥 ≥ 3} Ex: Find the domain of each of the following function. Express the answer in both set notation and inequality notation?