Introduction to Algebra

Transcription

1 INTRODUCTORY ALGEBRA Mini-Leture 1.1 Introdution to Alger Evlute lgeri expressions y sustitution. Trnslte phrses to lgeri expressions. 1. Evlute the expressions when =, =, nd = 6. ) d) Trnslte eh phrse to n lgeri expression, using x s the vrile. ) 6 less thn numer Fifty more thn numer The produt of 8 nd twie numer d) Thirteen less thn three times numer. Frnk hd $100 efore spending x dollrs on gift. How muh money remins? When numeril oeffiient is 1, the 1 is usully not written (e.g., 1x is usully written s x). Students often ignore term when they do not see oeffiient. Students often hve diffiulty trnslting phrses with less thn. Remind students tht expressions evlute to numer. Answers: 1) 9,,, d) ; ) x 6, x 50, 8 x, d) x 1; ) $100 x ML-6

2 Mini-Leture 1. INTRODUCTORY ALGEBRA The Rel Numers d e Stte the integer tht orresponds to rel-world sitution. Grph rtionl numers on the numer line. Convert from frtion nottion for rtionl numer to deiml nottion. Determine whih of two rel numers is greter nd indite whih, using < or >. Given n inequlity like, write nother inequlity with the sme mening. Determine whether n inequlity like 5 is true or flse. Find the solute vlue of rel numer. 1. Stte the integer tht orresponds to the sitution. ) Brent owes his prents $600. Jyne hs $15 in her heking ount.. Grph eh of the numers on the numer line: 1, 5,, Convert to deiml nottion. ) Use either < or > for to write true sentene. ) Write n inequlity with the sme mening. ) Write true or flse. ) Find the solute vlue. ) 6 Remind students tht integers re rtionl numers; ny integer n e written s the rtio of itself nd 1. Deiml numers tht terminte or repet in fixed lok re oth exmples of rtionl numers sk students to give exmples of oth. The deiml form of n irrtionl numer neither termintes nor repets. Some students hve never seen solute vlue efore nd will need exmples. Answers: 1) ; ) ; ) 075., 016,. 01. ; ) <, <, >; 5) 10 6, 5, 0 ; 6) True, flse, true; 7), 6, ML-7

3 INTRODUCTORY ALGEBRA Mini-Leture 1. Addition of Rel Numers Add rel numers without using the numer line. Find the opposite, or dditive inverse, of rel numer. Solve pplied prolems involving ddition of rel numers. 1. Add. ) e) f) d) g) Find the opposite, or dditive inverse. ) 6 5. Evlute x when: ) x 1 x A su diver is t depth of 16 ft elow the surfe. He desends nother 8 ft. Wht is his new depth? 5. On Jnury 1, in New Mrket, Indin, the temperture rose 17 F in three hours. If the strting temperture ws F, 5 wht ws the temperture three hours lter? Some students need to see ddition prolems done on the numer line first. Cution students out the differene etween the sutrt key nd the hnge-of-sign key on lultor. Refer students to the summry ox of rules for ddition of rel numers in the text. The numer line is good wy to illustrte opposite numers eh equidistnt from 0 ut on opposite sides of 0. Answers: 1) 1, 6, 0, d) 11., e) 1, f) 10, g) ) ft; 5) 1 F 10 ; ) 6, 5, 5 6 ; ) 1, 5; ML-8

4 Mini-Leture 1. Sutrtion of Rel Numers INTRODUCTORY ALGEBRA Sutrt rel numers nd simplify omintions of dditions nd sutrtions. Solve pplied prolems involving sutrtion of rel numers. 1. Rewrite the following s ddition prolems using the method of sutrtion s ddition of the opposite. ) Sutrt. ) Simplify. ) A su diver ws t depth of 17 ft elow the surfe. A wreked ship ws 1 ft lower thn the diver. Wht ws the depth of the wreked ship? 5. The Terre Hute Golf Clu showed profit of $7,000 one yer, while it hd loss of $19,000 in the next yer. Find the differene etween the mounts. Mny students find sutrting rel numers diffiult t first. Some students forget to hnge the sign of the seond numer fter hnging sutrtion to ddition. Enourge students to show this step (e.g., 5 ( 5) ). Mke sure students understnd the differene etween the symol s it reltes to sutrtion, negtive numer, or the opposite of numer. Answers: 1) 7, 6 ) 9 ft; 5) $91,000, 1 1 ; ), 10,. ; ), 5, 9 ; ML-9

5 INTRODUCTORY ALGEBRA Mini-Leture 1.5 Multiplition of Rel Numers Multiply rel numers. Solve pplied prolems involving multiplition of rel numers. 1. Multiply. ) d).. e) Multiply. ) d) 1 7. Evlute the expression when x. ) 5x 5x. Evlute the expression when x. ) x x 5. After diving 80 m elow se level, diver rises t rte of 6 m/min for 8 min. Where is the diver in reltion to the surfe t the end of the 8-min period? Refer students to the rules for multiplying rel numers in the text. Enourge students to look for ptterns to help them understnd the rules. Answers: 1) 0, 60, 150, d) 76., e) 15 ; ) 60, 0, 1, d) 1 ; ) 00, 80 ; ) 9, 9 ; 5) m ML-10

6 Mini-Leture 1.6 Division of Rel Numers INTRODUCTORY ALGEBRA d Divide integers. Find the reiprol of rel numer. Divide rel numers. Solve pplied prolems involving division of rel numers. 1. Divide, if possile. ) d) 0. Find the reiprol. 5 ) d) x. Rewrite eh division s multiplition. ) n 1 p. Divide, if possile. 15 ) d) 8 5. Lst yer, Perry s lss hd 50 students. This yer it hs 160 students. Find the hnge nd the perent inrese or perent derese from lst yer to this yer. Refer students to the rules for dividing rel numers in the text. Give exmples to show why division y zero is not defined ut zero n e divided y ny numer exept zero. Answers: 1),, not defined, d) 0; ) 1 5, 8, 1 6 n p; ), 7 5 6, 15, d) not defined; 5) 90, %, d) x ; ) 1, 5 8 5, ML-11

7 INTRODUCTORY ALGEBRA Mini-Leture 1.7 Properties of Rel Numers Find equivlent frtion expressions nd simplify frtion expressions. Use the ommuttive lws nd the ssoitive lws to find equivlent expressions. Use the distriutive lws to multiply expressions like 8 nd x y. d Use the distriutive lws to ftor expressions like x1 y. e Collet like terms. 1. Find n equivlent expression with the given denomintor. ) 7 7x x 6x. Simplify. 0x ) x Nme the lw (ommuttive, ssoitive, or distriutive) illustrted y eh sttement. ) d 5 5d d) List the terms of eh expression. ) x 6y x8y 1.5z 5. Multiply. 6 e) ) k 5h 5 6. Ftor. ) y 6x 1n x8y 1z 7. Collet like terms. ) 8x x 1n1m6n m.x.1y0.8x 1.8y Remind students tht the ommuttive lws del with the order of ddition or multiplition, wheres the ssoitive lws del with grouping. The distriutive lws over multiplition over ddition nd/or sutrtion. Hve students provide exmples to show whether or not the ommuttive/ssoitive lws hold for sutrtion nd division. Answers: 1) x 7x, 8 6x ; ) 5 6, ; ) ommuttive, ssoitive, distriutive, d) ommuttive, e) ommuttive; ) x, 6y, x, 8y, 1. 5z; 5) k 18, 5h 10, 10 ; n 1 xy z ; 7) 7x, 8n 15m, 15x. y. 6) y x,, ML-1

8 Mini-Leture 1.8 INTRODUCTORY ALGEBRA Simplifying Expressions; Order of Opertions d Find n equivlent expression for n opposite without prentheses, where n expression hs severl terms. Simplify expressions y removing prentheses nd olleting like terms. Simplify expressions with prentheses inside prentheses. Simplify expressions using the rules for order of opertions. 1. Find n equivlent expression without prentheses. 8x 5 y ) x8y z. Remove prentheses nd simplify eh expression. m 6 m ) f e 5f 1. Simplify. x5x1 ) x x. Simplify. ) d) 6 e) 1 f) 6 18 Remind students tht expressions n e simplified, wheres equtions re solved. Remind students tht n exponent pplies only to the expression just efore it (e.g., not 5x 5x). 5x mens 5x x, Answers: 1) 8x, 5 y, x8y z; ) m 1, 8, 7f 1e ; ) 5x, 60, x ; ) 10, 5, 15, d) 0, e), f) 6 5 ML-1