Official© Solutions Manual for Algebra & Trig,Larson,11e

C H A P T E R 1
Equations, Inequalities, and Mathematical Modeling

Section 1.1 Graphs of Equations .............................................................................51

Section 1.2 Linear Equations in One Variable .......................................................60

Section 1.3 Modeling with Linear Equations .........................................................68

Section 1.4 Quadratic Equations and Applications ................................................ 79

Section 1.5 Complex Numbers................................................................................94

Section 1.6 Other Types of Equations...................................................................100

Section 1.7 Linear Inequalities in One Variable ...................................................119

Section 1.8 Other Types of Inequalities ................................................................127

Review Exercises ........................................................................................................144

Problem Solving .........................................................................................................158

Practice Test .............................................................................................................162




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,C H A P T E R 1
Equations, Inequalities, and Mathematical Modeling
Section 1.1 Graphs of Equations
1. solution or solution point ?
11. (a) (1, 5): 5 = 4 − 1 − 2
2. graph ?
5 = 4 −1
3. intercepts 5 ≠ 3
4. y-axis No, the point is not on the graph.
?
5. origin (b) (6, 0): 0 = 4− 6− 2
?
6. numerical 0= 4 − 4
7. Two other approaches to solve problems mathematically 0 = 0
are algebraic and graphical. Yes, the point is on the graph.
8. Let ( x, y ) be any point on the circle. The distance ?

between the center ( h, k ) and ( x, y ) is the radius r. So,
12. (a) (6, 0): 2(6)2 2
+ 5(0) = 8
?
2 2
(x − h) + ( y − k ) = r . 2(36) + 5(0) = 8
?
? 72 + 0 = 8
9. (a) (2, 0): (2)2 − 3( 2) + 2 = 0 72 ≠ 8
?
4 − 6 + 2 = 0 No, the point is not on the graph.
?
0 = 0 (b) (0, 4): 2(0)2 2
+ 5( 4) = 8
Yes, the point is on the graph. ?
? 2(0) + 5(16) = 8
2
(b) (−2, 8): (−2) − 3( −2) + 2 = 8 ?
0 + 80 = 8
?
4 + 6 + 2= 8 80 ≠ 8
12 ≠ 8 No, the point is not on the graph.
No, the point is not on the graph.
13. y = −2 x + 5
?
10. (a) (0, 2): 2 = 0+ 4 x −1 0 1 2 5
2
2= 2
y 7 5 3 1 0
Yes, the point is on the graph.
?
( x, y ) (−1, 7) (0, 5) (1, 3) ( 2, 1) ( 52 , 0)
(b) (5, 3): 3 = 5+ 4
?
3 = 9
3 = 3
Yes, the point is on the graph.




© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. 51

,52 Chapter 1 Equations, Inequalities, and Mathematical Modeling


3 16. y = 5 − x 2
14. y = x −1
4
x −2 –1 0 1 2
−2 0 1 4 2
x 3
y 1 4 5 4 1
y − 52 –1 − 14 0 1
2
x, y (−2, 1) (−1, 4) (0, 5) (1, 4) (2, 1)
( x, y ) (−2, − 52 ) (0, −1) (1, − 14 ) ( 43 , 0) (2, 12 )




17. x-intercept: ( − 2, 0)
2
15. y = x − 3x
y-intercept: (0, 2)

x −1 0 1 2 3
18. x-intercept: ( 4, 0)
y 4 0 –2 –2 0
y-intercepts: (0, ± 2)
( x, y ) (−1, 4) (0, 0) (1, − 2) ( 2, − 2) (3, 0)
19. x-intercept: (3, 0)

y-intercept: (0, 9)

20. x-intercepts: ( ± 2, 0)

y-intercept: (0, 16)

21. x-intercept: (1, 0)

y-intercept: (0, 2)

22. x-intercepts: (0, 0), ( ± 2, 0)

y-intercept: (0, 0)

23. x 2 − y = 0
2
(− x) − y = 0  x 2 − y = 0  y -axis symmetry
x 2 − ( − y ) = 0  x 2 + y = 0  No x-axis symmetry
2
(− x) − ( − y ) = 0  x 2 + y = 0  No origin symmetry

24. x − y 2 = 0

(− x) − y 2 = 0  − x − y 2 = 0  No y -axis symmetry
2
x − ( − y ) = 0  x − y 2 = 0  x-axis symmetry
2
(− x) − (− y ) = 0  − x − y 2 = 0  No origin symmetry




© 2022 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

, Section 1.1 Graphs of Equations 53


25. y = x3
3
y = ( − x)  y = − x 3  No y -axis symmetry
− y = x3  y = − x 3  No x-axis symmetry
3
− y = ( − x)  − y = − x 3  y = x 3  Origin symmetry

26. y = x 4 − x 2 + 3
4 2
y = ( − x) − ( − x ) + 3  y = x 4 − x 2 + 3  y -axis symmetry
− y = x 4 − x 2 + 3  y = − x 4 + x 2 − 3  No x-axis symmetry
4 2
− y = ( − x) − ( − x ) + 3  y = − x 4 + x 2 − 3  No origin symmetry

x
27. y =
x2 + 1
−x −x
y = 2
 y = 2
 No y -axis symmetry
( − x) +1 x +1
x −x
−y =  y = 2  No x-axis symmetry
x2 + 1 x +1
−x −x x
−y = 2
 −y = 2  y = 2  Origin symmetry
(− x) + 1 x + 1 x +1

1
28. y =
1 + x2
1 1
y = 2
 y =  y -axis symmetry
1 + (− x) 1 + x2
1 −1
−y = 2
 y =  No x-axis symmetry
1+ x 1 + x2
1 −1
−y = 2
 y =  No origin symmetry
1 + ( − x) 1 + x2

29. xy 2 + 10 = 0
(− x) y 2 + 10 = 0  − xy 2 + 10 = 0  No y -axis symmetry
2
x( − y ) + 10 = 0  xy 2 + 10 = 0  x-axis symmetry
2
(− x)(− y ) + 10 = 0  − xy 2 + 10 = 0  No origin symmetry

30. xy = 4
(− x) y = 4  xy = −4  No y -axis symmetry
x( − y ) = 4  xy = −4  No x-axis symmetry
(− x)(− y ) = 4  xy = 4  Origin symmetry
31. 32.




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