Lesson 5.1 Exercises, pages
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1 Lesson 5.1 Eercises, pages A 4. Use the given graphs to write the solutions of the corresponding quadratic inequalities. a) < 0 The solution is the values of for which y<0; that is, 1<<5, ç b) The solution is the values of for which y» 0; that is, 1 or» 5, ç y y c) > 0 The solution is the values of for which y>0; that is, 2<<8, ç d) The solution is the values of for which y 0; that is, 2 or» 8, ç y y Solve this quadratic inequality. ( - 2)( - 6) 0 Solve: ( 2)( 6) 0 2 or 6 When 2, such as 0, L.S. 12; R.S. 0; so 0 does not satisfy the inequality. When 2 6, such as 4, L.S. 4; R.S. 0; so 4 does satisfy the inequality. The solution is: 2 6, ç 1
2 B 6. Solve each quadratic inequality. Represent each solution on a number line. a) Solve: ( 4)( 3) 0 4 or 3 When 3, such as 4, L.S. 8; R.S. 0; so 4 does not satisfy the inequality. When 3 4, such as 0, L.S. 12; R.S. 0; The solution is: 3 4, ç b) > 0 Solve: (2 3)(2 1) or 0.5 When < 1.5, such as 2, L.S. 3; R.S. 0; so 2 does satisfy the inequality. When > 0.5, such as 0, L.S. 3; R.S. 0; The solution is: < 1.5 or > 0.5, ç c) Solve: (2 1)( 3) or 3 When 0.5, such as 1, L.S. 4; R.S. 0; so 1 does not satisfy the inequality. When 0.5 3, such as 0, L.S. 3; R.S. 0; The solution is: 0.5 3, ç DO NOT COPY. P
3 7. Solve each quadratic inequality. Represent each solution on a number line. a) -5 2 > Solve: (5 3)( 4) or 4 When < 4, such as 5, L.S. 125; R.S. 97; so 5 does not satisfy the inequality. When 4<<0.6, such as 0, L.S. 0; R.S. 12; The solution is: 4<<0.6, ç b) > -14 Solve: ( 2)(4 7) 0 2 or 1.75 When < 2, such as 3, L.S. 9; R.S. 14; so 3 does satisfy the inequality. When > 1.75, such as 0, L.S. 0; R.S. 14; The solution is: < 2 or > 1.75, ç Solve each quadratic inequality by graphing. Give the solutions to the nearest tenth. a) b) 0 < Rearrange the inequality. Graph: y The critical values are Graph: y approimately 1.1 and The critical values are The solution of the inequality approimately 3.9 and 1.0. is the values of for which The solution of the inequality y>0; that is, is these points and the values 1.1<<32.6, ç. of for which y 0; that is, , ç. 3
4 9. Use the quadratic formula to solve each quadratic inequality. Give the solutions to the nearest tenth. a) b) < 0 5 > 1 Solve: Substitute: a 2, b 3, c 4 Solve: Substitute: a 5, b 6, c 15 in: b_ b 2 4ac 2a 3_ ( 3) 2 4(2)( 4) 2(2) 6_ 336 3_ or or 1.2 When < 0.9, such as 1, When < 2.4, such as 3, L.S. 1; R.S. 0; so 1 L.S. 1.8; R.S. 1; so 3 does not satisfy the inequality. does satisfy the inequality. When 0.9<<2.4, such as When >1.2, such as 3, 0, L.S. 4; R.S. 0; L.S. 4.2; R.S. 1; so 3 so 0 does satisfy the does satisfy the inequality. inequality. The solution is: The solution is: < 2.4 or >1.2, ç 0.9<<2.4, ç in: b_ b 2 4ac 2a 6_ 6 2 4(5)( 15) 2(5) 10. Solve each quadratic inequality. Give the solutions to the nearest tenth where necessary. Use a different strategy each time. Eplain each strategy. a) 3 2 < 21 Use intervals and test points. Solve: 3 2 < ( 7) 0 0 or 7 When <0, such as 1, L.S. 3; R.S. 21; so 1 does not satisfy the inequality. When 0<<7, such as 1, L.S. 3; R.S. 21; so 1 does satisfy the inequality. The solution is: 0<<7, ç b) > Rearrange the inequality >0 Use a graphing calculator. Graph: y The critical values are approimately 2.2 and 3.0. The solution of the inequality is the values of for which y>0; that is, < 2.2 or >3.0, ç. 4 DO NOT COPY. P
5 11. Consider this inequality: a) Solve the inequality by factoring. Illustrate the solution on a number line. Solve: (2 5)(2 5) When 2.5, such as 2, L.S. 1; R.S. 0; so 2 does not satisfy the inequality. When» 2.5, such as 3, L.S. 1; R.S. 0; so 3 does not satisfy the inequality. The solution is: b) Solve the inequality by graphing. Sketch the graph. y y The graph of y opens up, has -intercept 2.5 and is congruent to y 4 2. The only point where is at c) i) What do you notice about the solution of the inequality? The solution is one number. ii) Write a different inequality that has the same number of solutions. What strategy did you use? Sample response: Write the equation of a parabola that touches the -ais; for eample, y ( 3) 2. Another inequality with eactly one solution is: 2 6 9» The product of two consecutive even integers is at least 48. What might the integers be? Let the lesser integer be, then the greater integer is 2. An inequality is: ( 2)» 48 Solve the equation: ( 6)( 8) 0 6 or 8 When 6, 2 8 When 8, 2 6 Any pair of consecutive even integers greater than or equal to 6 and 8, or less than or equal to 8 and 6 are solutions to the problem; such as 8 and 10, 10 and 12, 8 and 10, 10 and 12. 5
6 13. A tennis ball is thrown upward at an initial speed of 15 m/s. The approimate height of the ball, h metres, after t seconds, is given by the equation h = 15t - 5t 2. Determine the time period for which the ball is higher than 10 m. An inequality that represents this situation is: 15t 5t 2 >10 A related quadratic equation is: 5t 2 15t 10 0, or t 2 3t 2 0 Solve the equation. (t 1)(t 2) 0 t 1 or t 2 When t<1, such as t 0, L.S. 0; R.S. 10; so t 0 does not satisfy the inequality. When 1<t<2, such as t 1.5, L.S ; R.S. 10; so t 1.5 does satisfy the inequality. The solution is: 1<t<2 So, the tennis ball is higher than 10 m between 1 and 2 s after it is thrown b + 7 = 0 is a quadratic equation. For which values of b does the quadratic equation have: a) two real roots? For 2 2 b 7 0 to have 2 real roots, its discriminant is greater than 0. b 2 4ac> 0 Substitute: a 2, c 7 b 2 4(2)(7)>0 b 2 >56 b> 56 or b< 56 b) no real roots? For 2 2 b 7 0 to have no real roots, its discriminant is negative. b 2 4(2)(7)<0 b 2 <56 b< 56 and b> 56, which is written 56<b< 56 6 DO NOT COPY. P
7 C 15. Create an inequality that has each solution. a) b) < -1.1 or > 0.4 Work backward. The critical points are The critical points are 1.1 and and 3. A related quadratic So a related quadratic equation is: equation is: ( 1.1)( 0.4) 0 ( 13)( 3) On a graphing calculator, On a graphing calculator, graph the related function. graph the related function. The solution is all real The solution is the -intercepts numbers to the left and to and all real numbers between the right of the -intercepts. them. An inequality is: An inequality is: >0 7
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