Quadratic and Rational Inequalities

Transcription

1 Chapter Qadratic Eqations and Ineqalities. Gidelines for solving word problems: (a) Write a verbal model that will describe what yo need to know. (b) Assign labels to each part of the verbal model nmbers to the known qantities and letters to the variable qantities. (c) Use the labels to write an algebraic model based on the verbal model. (d) Solve the reslting algebraic eqation and check yor soltion.. Unit Analysis 9 dollars hors 18 dollars hor 7. An eample of a qadratic eqation that has only one repeated soltion is. Any eqation of the form c, where c is a constant will have only one repeated soltion. Section. Qadratic and Rational Ineqalities ± 9 Critical nmbers, 9, 9, Critical nmbers, 1 11.,,, <, 1 > 1.,,, 1 >, < 1.,,,, >, 1 11 <, 1 >

2 Section. Qadratic and Rational Ineqalities ,,, 1, 1,,, <, >, < 1, > 1, 1 <, 1 7 > 1.,,, 1. <. 8 8,, 8 8, <,,,,, > 1. 1, > >,,,,,,,,,,,,, 1 1,,. > 1 > 1 >, 1,, 1 1,, 1,. < ± 1 No critical nmbers is not less than zero for any vale of. none for all real nmbers,

3 Chapter Qadratic Eqations and Ineqalities 9. > ± 1 8 ± 8 ± ± 1. 9 for all real nmbers 1 1,,,,,, <. <,, none 8,,,,, > 9. 7 > 19 1 < Mltiply by 1 1 > <,,,,, 1 1,,, 1,, 1 1, 1

4 Section. Qadratic and Rational Ineqalities > for all real nmbers ecept , 7 7, none < 7. 1 < ±, 1 19 > 1 ± ± 1 ±,,,,, , 1, 9 9, 1 1,, 9 1, > 1. Keystrokes:,, Y X,T, X,T,,, 1,, 8,,, 1 1 1

5 Chapter Qadratic Eqations and Ineqalities. Keystrokes:. Keystrokes: Y. X,T, 1. X,T, Y X,T, X,T,,, y 1 y 9, 1, ENTER Keystrokes: 1 y 1 Y 9. X,T, y, 7, , > 7. >,,,,,, <,,,,, 79.,,,,,

6 Section. Qadratic and Rational Ineqalities 81. y y < 8.,,,, y, 8 y y y y, 11,, 11 11,, 11,,,,,, 11 8 y < 89. 1, 1, 1 1,, 1, 1 1 > > 8 1 > 7 > > >,, 7 7,, 7 7 7, < < < < 1 9.,,,,, ,,,,, 1, 1

7 Chapter Qadratic Eqations and Ineqalities 9. Keystrokes: Y 1 X,T, X,T,, 1, Keystrokes: Y X,T, X,T, 1, 1, Keystrokes: 11. Keystrokes: Y X,T, X,T, ENTER Y X,T, 1 X,T, y 1 y,. y 1 ENTER y,.8.18, Keystrokes: 1. Keystrokes: Y X,T, X,T, Y X,T, X,T, (a) Soltion, (a),, Look at -ais and vertical asymptote (b), (Graph y as and find the intersection.) y (Graph y 1 as and find the intersection.) (b) Soltion, (Notice graph stays below line y. ) y height > 1t 18t > 1t 18t > t 8t 1 < t t <,,,,,

8 Section. Qadratic and Rational Ineqalities r r cannot be negative. 18, 11 r > r r > 11 1 r 1r > 11 1r r 1 > r r > ,,,.7,, r > 7.% Verbal model: Profit > 1,, Revene Cost Profit > 1,,. 1 1, > 1,,. 1 1, > 1,,. 8 1, > 1,, 9,, 1,, 9, 9,, 1, 1,, 9,, 1, 9, 1, nits >. 8 1,8, >. 9, 11. Area > l l > l l > l l > l l < l l 1 < l, 1, 1 1,, 1, 11. (a) Keystrokes: Y. 1. X,T, 1.1 X,T,. X,T, (b) Let y and find the intersection of the graphs..7, 1.7,.7 t

9 8 Chapter Qadratic Eqations and Ineqalities 117. The direction of the ineqality is reversed, when both sides are mltiplied by a negative real nmber A polynomial can change signs only at the -vales that make the polynomial zero. The zeros of the polynomial are called the ciritical nmbers, and they are sed ro determine the test intervals in solving polynomial ineqalities < is one eample of a qadratic ineqality that has no real soltion. Any ineqality of the form c <, c any positive constant or c >, c any positive constant will not have a real soltion. Review Eercises for Chapter y 1. y y 1 y 1y 1 y y 1 y 1 y 1 y y 1 y 1 y 1 y y y y , ± 1 9 ± y z 11 y 1 1 ± z ±11 y ±1 1 ± z ±11i y ±, 19. y 1. y y ± y ±i y 18 y 18 y ±18 y ± i ±1 ± ±i Check: Check: 9 9? 1 1? 9 1? 1?