Idiot's Guides: Algebra I (2015)
APPENDIX B. Check Point Answers
Chapter 1
1. is b) a rational number and d) a real number.
2. so it is a) an integer, b) a rational number, and d) a real number.
3. is c) an irrational number and d) a real number.
4. so it is b) a rational number and d) a real number.
5. distributive property
6. inverse (for addition)
7. commutative property (for addition)
8. closure
9. identity (for multiplication)
10. associative property (for multiplication)
11. 4 + − 17 + − 3 + 29 = 4 + − 20 + 29 = 4 + 9 = 13
12. −7·5·−8·12 = −7·−8·5·12 = 56·60 = 3,360
13. 8·−6·3·−4 = 8·3·−6·−4 = 24·24 = 576
14. − 14 + 8 + − 31 + 27 + 6 + − 4 = − 8
15. − 25 (18 + − 14) + 8(− 9 + 12) = − 25 (4) + 8(3) = − 100 + 24 = − 76
16. 7(− 39 + 43) + 14·− 2 = 7(4) + 14·− 2 = 28 + − 28 = 0
17.
18. 4·(− 0.6)·(l.2)·(− 10)2 =− 288
19. −1.4 + 8.3 + − 3.1 + 2.7 + − 4.5 = 2
20.
21. (23)·(22) = 25 = 32
22.
23. (32)3 = 36 = 729
24.
25.
26. 6−3·65 = 62 = 36
27.
28.
29.
30.
31. 4 − 3·4 + 3 = 4 − 12 + 3 = − 8 + 3 = − 5
32. (4 − 3)·(4 + 3) = 1·7 = 7
33. 4 − 3·(4 + 3) = 4− 3·7 = 4 − 21 = − 17
34. (4 − 3)·4 + 3 = 1·4 + 3 = 4 + 3 = 7
35. 52 − 2·(7 − 3) = 25 − 8 = 17
36. (66 − 54) ÷ 3 + 10 ÷ 5 − (6 − 22) = 4 + 2 − 2 = 4
37. 5 − 2 + 2·(8 − 5) ÷ 2 = 3 + 3 = 6
38. 17 − (8 − 3)·(2 + 1) ÷ 5 = 14
39. 8(2(4 + 3)2 − 20 + 12) ÷ 4 = 720 ÷ 4 = 180
40. 7 + 32 ÷ 2 − 6 = 5.5
Chapter 2
1. −4x
2. y + 18
3.
4. 9x − 3
5. 19 + (− 3)x = 19 − 3x
6. (a − b)· − 5 = −5(a − b)
7. 19 − x
8. 2x + 4 = 8 + y
9. 3h + 1.50 = 12
10. n + (n + 15) = 42
11. 2 factors: 3 and x
12. 3 factors: -6, y and y
13. 3 factors: 23, x and y
14. 6 factors: -48, x, x, x, y, and y
15. 3 terms
16. 2 terms
17. 1 term
18. 2 terms
19. 3 terms
20. 5 terms
21. (x + 3) + (9 + x) = 2x + 12
22. (2x − 1) + (3x + 9) = 5x + 8
23. (x2 + 5) + (x2 + 6x) + (8x + 3) = 2x2 + 14x + 8
24. (2x + 7) + (8x + 4) + (− x2 − 1) = −x2 + 10x + 10
25. (2x − 4) + (6x + 1) + (3 − x) = 7x
26. (y + 5) + (2y − 9) + (4y + 10) + (y − 6) = 8y
27. (8t − 4) + (− t − 1) + (13t − 7) + (− 2t − 9) = 18t − 21
28. (x2 + 4) + (− x2 − 1) + (−3 − 5x) + (10x + 9) = 5x + 9
29. (3x2 + 2x + 4) + (x2 − 3x + 10) = 4x2 − x + 14
30. (4x2 − 5x − 6) + (9 + 7x − 4x2) = 2x + 3
31. 2(x − 4) + 7(x + 4) = 2x − 8 + 7x + 28 = 9x + 20
32. −3(y + 8) + 8(y − 3) = −3y − 24 + 8y − 24 = 5y − 48
33. 6(2t + 5) + 9(t − 2) = 12t + 30 + 9t − 18 = 21t + 12
34. −7(3w − 5) + 4(w + 5) = − 21w + 35 + 4w + 20 = − 17w + 55
35. x(x + 4) − (x + 4) = x2 + 4x − x − 4 = x2 + 3x − 4
36. −8x(x − 3) + 6(x − 4) = −8x2 + 24x + 6x − 24 = −8x2 + 30x − 24
37. 5a(2a − 7) − 2(8 − a) = 10a2 − 35a − 16 + 2a = 10a2 − 33a − 16
38. −4y (7 − 3y) − 4(3 − 7y) = −28y + 12y2 − 12 + 28y = 12y2 − 12
39. 3x (2x + 3y) + 3y (2x + 3y) = 6x2 + 9xy + 6xy + 9y2 = 6x2 + 15xy + 9y2
40. 2x2 (x2 − 7x + 9) + 3x (x2 − 7x + 9) + 2 (x2 − 7x + 9) = = 2x4 − 11x3 − x2 + 13x + 18
41. 2x2 + x − 28
42. x3 − 2x2 − x − 6
43. 2x3 + 15x2 + 13x + 3
44. 6x3 − 17x2 − 12x + 32
45. (x + 3)·(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6
46. (x + 4)·(x − 2) = x2 − 2x + 4x − 8 = x2 + 2x − 8
47. (2x + 1)·(x + 7) = 2 x2 + 14x + x + 7 = 2x2 + 15x + 7
48. (4x − 1)·(2x + 3) = 8x2 + 12x − 2x − 3 = 8x2 + 10x − 3
49. (5x − 2)·(2x − 3) = 10x2 − 15x − 4x + 6 = 10x2 − 19x + 6
50. (x2 + x)·(x − 3) = x3 − 3x2 + x2 − 3x = x3 − 2x2 − 3x
Chapter 3
1. Function. No rule.
2. Function. It’s hard to see but this does follow a rule. Multiply the input by -3 and then add 4.
3. Function. Output = Input squared, minus 1.
4. Not a function. No rule.
5. Function. The outputs are the positive square roots of the inputs.
6. Domain: {−2, −1,0,1,2}, Range: {3,4,5,6,7}, Rule: Output = input + 5
7. Domain: {−2, −1,0,1,2}, Range: {4}, Rule: Output always equals 4, no matter what the input.
8. Domain: {0,2,4,6,8}, Range: {−3,1,5,9,13}, Rule: Output = twice the input minus 3.
9. Domain: {10,20,30,40,50}, Range:
{1,11,21,31,41}, Rule: Output = input minus 9.
10. Domain: {−5,6,8,11,21}, Range:
{−4,−1,−7,12,−5}, Rule: None
11. f (3) = 4 − 3·3 = 4 − 9 = − 5
12. f (−5) = 4 − 3(−5) = 4 + 15 = 19
13. f (0) = 4 − 3(0) = 4 − 0 = 4
14. f (−1) = 4 − 3 (−1) = 4 + 3 = 7
15. f (−100) = 4 − 3 (−100) = 4 + 300 = 304
16. If g(x) = 8x − 2 find
g (−4) = 8 (−4) − 2 = − 32 − 2 = − 34
17. If h(x) = x2 + 5 find h(0) = 02 + 5 = 0 + 5 = 5
18. If (x) = x2 − 3x + 7 find
p(−1) = (−1)2 − 3(−1) + 7 = 1 + 3 + 7 = 11
19. If v(t) = 25 − 8t find
v(5) = 25 − 8·5 = 25 − 40 = − 15
20. If a(t) = 9.8 find a(20) = 9.8 (Constant function)
21.
22.
23.
24.
25.
26. v(t) = 5t − 3
27. f (x) = − 3x + 7
28. g(x) = −x2 + 5
29. a(t) = 1.5t + 2
30. p(x) = 3x + 10
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41. function
42. not a function
43. function
44. not a function
45. function
46. not a function
47. function
48. function
49. function
50. not a function
Chapter 4
1. 3x + 3 = 132
2. 3y + 3 = y + 3
3. 2t − 10 = 5 + t
4. 2x − 8 = 2x − 1
5. −3a + 23 = 6a − 14
6. 2x + 3 = 5x + 3
7. 4v − 12 = − 2v + 4
8. 4x − 17 = 9x + 23
9. 6t − 15 = 6t − 15
10. −3a − 19 = 4 − 3a
11. x = 7
12. t = 20
13. y = 27
14. 48 = t
15. a = −25
16. w = 11
17. x = −9
18. t = 349
19. y = −120
20. z = 0
21. a = 7
22. z = −4
23. n = −7
24. y = 1
25. t = −5
26. x = 80
27. z = 2.5
28. w = −4
29. x = 3
30. y = −217
31. − 4 = x
32. 7 = t
33. y = 12
34. v = 45
35. 7 = x
36. no solution
37. 53 = x
38. identity
39. z = 0
40. −11.5 = x
41. The number is 7.
42. The number is 4.
43. The two consecutive numbers are 26 and 27.
44. 5,000 storage containers were manufactured.
45. The garden measures 6 feet by 9 feet.
46. The price of a hot dog is $3.50.
47. Carlos and Dahlia will have the same amount in savings in 10 weeks.
48. The average price per e-book was $7.89.
49. Sophie bought 6 pounds of almonds.
50. The number is 8.
Chapter 5
1. x > 5
2. r ≤ 4% or r ≤ 0.04
3. h > 48
4. p ≥ 20
5. I ≤ 12,000
Sample answers for problems 6 through 10.
6. A number, x, is at least 16.
7. A number, y, is less than 12.
8. A number, t, exceeds 22.
9. A number, z, is no more than 100.
10. A number, a, is more than 0.
11. x > − 3
12. y < 6
13. x ≥ 4
14. t > − 4.5
15. y ≤ 1.5
16. x = − 4 is part of the solution
17. x = − 6 is not part of the solution
18. y = 1 is not part of the solution
19. x = 5 is not part of the solution
20. t = 10 is part of the solution
21. x > 5
22. t ≥ −8
23. y < − 4
24. z ≥ 7
25. a ≤ 9
26. x ≤ 4
27. −2 > x
28. x ≤ − 4
29. y > − 1
30. t ≤ 0
31. y > 3 or y < − 2
32. 1 ≤ z ≤ 3
33. 18 ≤ x or x < 7
34. 3 < t < 4
35. 17 < 4y + 5 ≤ 1 has no solution.
36. −8 ≤ t or t > − 5 is equivalent to t ≤ −8
37. −4 ≤ x ≤ 6
38. 5 < x or x < 3.5
39. −4 > t ≥ −5
40. 2 ≤ y < 4
Chapter 6
Answers for problems 1 through 5 are shown on the graph.
6. (4, 1) or (8,9)
7. (0, 4) or (0, -3)
8. (3, -1) or (7, -5)
9. (-6, -3) or (-8, -9)
10. (-5, 0) or (2, 0)
Sample tables are shown for problems 11 through 20.
11. f (x) = 3x − 4
12.
13. y = 2x − 3
14. g(x) = x + 5
15. y = −x + 5
16. f (x) = −2x + 7
17.
18. y = 4 − 3x
19. f (x = −2x)
20.
21. x-intercept: (3, 0), y-intercept: (0, -6)
22. x-intercept: (2, 0), y-intercept: (0, 6)
23. x-intercept: (-1, 0), y-intercept: (0, 4)
24. x-intercept: (5, 0), y-intercept: (0, 5)
25. x-intercept: (-4, 0), y-intercept: (0, 4)
26. x-intercept: (4, 0), y-intercept: (0, 8)
27. x-intercept: (2, 0), y-intercept: (0, -6)
28. x-intercept: (6, 0), y-intercept: (0, -6)
29. x-intercept: (-5, 0), y-intercept: (0, 5)
30. x-intercept: (8, 0), y-intercept: (0, -4)
31. Slope: , y-intercept: (0, -5)
32. Slope: , y-intercept: (0, -7)
33. Slope: , y-intercept: (0, 9)
34. Slope: , y-intercept: (0, -3)
35. Slope: , y-intercept: (0, -4)
36. Slope: , y-intercept: (0, 7)
37. Slope: , y-intercept: (0, 5)
38. Slope: , y-intercept: (0, -9)
39. Slope: , y-intercept: (0, 3)
40. Slope: , y-intercept: (0, 1)
41. direct variation
42. No, direct variation doesn’t have constant term.
43. No, xy = 12 is equivalent to . Direct variation cannot have a variable in the denominator.
44. direct variation
45. No, the graph of a direct variation equation passes through the origin.
46. k = 45
47. k = 2π
48. The equation is F = 50a.
49. Two pounds of potatoes cost $2.98.
50. At 30 degrees Celsius, the volume will be 1,800 cubic centimeters.
Chapter 7
1.
2.
3.
4. y = 3x − 9
5. y = − 4x + 2
6.
7. y = −6x + 3
8.
9.
10. y = 10x + 10
11. 6x − y = 7
12. 3x − y = − 13
13. 3x + y = 5
14. 4x + y = 11
15. 8x − 4y = 32
16. x − 2y = 2
17. 4x + 5 y = 3
18. 6x + 2y = − 19
19. 3x − 2y = 7
20. 3x + 5y = 28
21. y − 7 = −2(x + 5) point-slope form
22. 3x − 4y = 12 standard form
23. y = 2x − 6 slope-intercept form
24. x + y = 10 standard form
25. point-slope form
26. y = − 3x + 0 or simply y = −3x
27. y = 5x − 9
28.
29.
30. y = 1x + 3 or simply y = x + 3
31. y = −4x + 3
32. y = 5x + 16
33.
34.
35. y = − x + 5
36. y = 2x − 3
37.
38. y = x + 1
39.
40. y = 3x − 23
41.
42.
43.
44.
45.
46. , or y + 10 = − 3 (x − 5)
or y − 20 = − 3 (x + 5)
47. , or
48. , y + 5 = −4(x − 4) or y − 7 = − 4 (x − 1)
49. , or
50.
51. Use 2. y + 3 = 2 (x − 6) or y = 2x − 15.
52. . Use . .
53. Use m = 4. y + 3 = 4(x + 4) or y = 4x + 13.
54. Use . or .
55. Use .
56. Use . or .
57. Use .
58. . Use m = -2. y − 7 = −2(x + 1) or y = −2x + 5.
59. Use m = -3. or
.
60. Use m = -1. y + 3 = −1(x − 4) or y = −x + 1.
Chapter 8
1. |9 − n| = 6 or |n − 9 = 6
2. |t − (−4)| = 10 or |−4 − t| = 10
3. |x − 5| = 12 or |5 − x| = 12
4. 2|y − (− 4)| = 12 or 2| − 4 − y| = 12
5. |z − 12| = z + 4 or |z − 12| = z + 4
Sample sentences are given for problems 6 through 10.
6. The number 7 is 3 units from x.
7. The distance between t and -2 is 9 units.
8. The number y is 11 units from 8.
9. Five times the distance between x and 4 is 20.
10. The distance between z and 7 is 3 less than z.
11. x = 3 or
12. t = − 1 or t = −4
13. y = 9 or y = 7
14. x = 8 or x = 18
15. |11 + 8x| = − 17 has no solution. Absolute value is never negative.
16. or x = −2
17. y = 3 or y = 1
18. x = 2 or x = −4
19. or x = 2
20. or 1 = x
21. or x > 2
22.
23. −2.8 ≤ y ≤ 2
24. or z ≥ 2
25. −2.75 ≤ a ≤ 2
26. 6.5 > x > − 4
27. −1.75 ≥ p or p ≥ − 1
28. |9t + 5| ≤ −2 has no solution. Absolute value is never negative.
29. −21 < x < 11
30. |7y − 23| > 0 is true for all values of the variable. The solution set is all real numbers.
31. x = 36 or x = − 8
32. y = − 41 or y = 7
33. The number n is between 6 and 12, inclusive.
34. Paul’s salary is between $86,500 and $78,500.
35. The points are (0, -10) or (0, 2).
36. The possible points are (-25, 0) or (1, 0).
37. The points are (0, -7) or (0, 13).
38. 96.6 > t or t > 100.6
39. The range of acceptable voltages is 100 < V < 120.
40. The candidate should receive between 42% and 50% of the vote.
41. y = |x − 3|
42. y = |x| + 1
43. y = |x + 1|
44. y = |x| − 5
45. y = |2x|
46. y = −3|x|
47. y = |2x − 5|
48. y = |7 − 4x|
49. y = |3x − 8|
50. y = |12 − 5x|
Chapter 9
1. y = 12 Horizontal line
2. x = 5y Oblique line
3. y = 12 + x Oblique line
4. x = 7 Vertical line
5. x − 6 = 0 Vertical line
6. x = − 4
7. y = − 3
8. x + 4 = 6
9. y + 7 = 4
10. x = 1
11. y ≤ 2x − 1
12.
13. y < −3x + 5
14.
15. x < 3
16. y − 5 ≥ 0
17. x + y ≤ 8
18. 2x − y > − 6
19. 4x − 3y < 12
20. x ≤ 2y + 8
21. y < |x − 7|
22. y ≥ |x + 3|
23. y > |x| − 4
24. y ≤ |x| + 1
25.
26. y ≥ |2x| − 3
27.
28.
29. y ≤ −2|x| + 5
30. y > −3|x + 4| + 5
Chapter 10
1. y = 2x + 3
y = 3x − 1
2.
3. x + y = 9
x − y = 1
4. 3x − y = 5
6x − 2y = 8
5. 2x − 5y = −44
y = 6 − x
6. y = 3x − 7
2x − 3y = 14
7.
8. 2x + y = −8
3x − 5y = 14
9. x + y = 6
y= − x − 4
10. 3x − 2y = 5
x + 4 y = −3
11. x = 1, y = 10
12. , y = −2
13. x = 3, y = −3
14. x = −2, y = −5
15. a = 1, b = 6
16. x = 0, y = 9
17. x = 5, y = 2
18. x = 7, y = 3
19. x = −1, y = 2
20. a = −1, b = 2
21. x = 10, y = 7
22. x = 7, y = 1
23. x = 2, y = 3
24. x = 3, y = 0
25. x = 3, y = −3
26. x = 4, y = −3
27. x = 7, y = 3
28. x = 5, y = 2
29. x = 3, y = 4
30. x = −1, y = 2
31. There are 7 quarters and 13 dimes.
32. There were 18 of the 14 cent stamps and 12 of the 17 cent stamps.
33. Use 20 pounds of $6 coffee and 40 pounds of $5.10 coffee.
34. Invest $2,600 at 4% and $3,400 at 7%.
35. Use 16 ml of 5% solution and 32 ml of 20% solution.
36. She traveled 6.4 hours at 50 mph and 1.6 hours at 20 mph.
37. Adult admission was $10 and children’s admission was $3.50.
38. There were 7 nickels and 13 dimes.
39. Pens cost 80 cents and pencils cost 30 cents.
40. There were 110 adults and 140 children.
41. y ≤ 2x + 1
y ≥ 4 − x
42. y < 7 − 3x
2x + y > 8
43. 3x − y < 1
y ≥ 2 − 3x
44. 5x + 2y ≥ 12
3x + 7 < y
45. 3x + 4 ≥ 4y
x + 2y < 12
46. x + y ≥ 40
x − y ≤ 8
47. x − y ≤ 12
x + y ≥ 20
48. x + y ≤ 80
100x + 300y ≥ 7500
49. 15x + 20y ≤ 400
12x + 30y ≤ 300
50. 2x + 9y ≤ 25
3x + 4y ≤ 36
Chapter 11
1. is not a polynomial.
Division by a variable is not permitted, because they would be represented by negative exponents.
2. 3y7 − 4y3 + 8 is a polynomial.
3. is not a polynomial because there is a variable under a radical. Radicals are the equivalent of fractional exponents.
4.
5.
6.
7. is a binomial of degree 1.
8. is a trinomial of degree 2.
9. is a monomial of degree 3.
10. is a polynomial of degree 12.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
Chapter 12
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
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20.
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30.
31.
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35.
36.
37.
38.
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41.
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44.
45.
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47.
48.
49.
50.
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52.
53.
54.
55.
56.
57.
58.
59.
60.
Chapter 13
1.
2.
3.
4.
5.
6.
7.
8. is prime.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25. is prime.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
Chapter 14
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
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13.
14.
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17.
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Chapter 15
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16. No solution in the real numbers.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26. There are two rational solutions.
27. There are two irrational solutions.
28. There are no real solutions.
29. There is one solution.
30. There are two rational solutions.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
Chapter 16
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36. Vertex (2, 3)
37. Vertex (5, −3)
38. Vertex (1, −3)
39. Vertex (−4, 8)
40. Vertex (5,−4)
41. It will take 2.646 seconds to hit the ground.
42. feet high after one second.
43. The object is 76 feet high after 1.5 seconds.
44. After 2 seconds, the object is 261 feet high.
45. The maximum height the object reaches is 405 feet after 5 seconds.
46. There are two moments when the object is 81 feet off the ground, half a second after
it is thrown, and again on the way down, 9.5 seconds after it was thrown.
47. The object hits the ground just after 10 seconds.
48. After 0.5 seconds, the object is 66.25 meters from the ground.
49. The object is 20 meters high at approximately 2.14 seconds after it was released.
50. It takes approximately 2.65 seconds to hit the ground.
Chapter 17
1. All real numbers except 0.
2. All real numbers except 7.
3. All real numbers except -8.
4. All real numbers except 3.
5. All real numbers except 1 and -1.
6. All real numbers except -2.
7. All real numbers except -3.
8. All real numbers except 0 and -5.
9. All real numbers except -2 and -3.
10. All real numbers. (There is no value of x that can make the denominator equal 0.)
11.
12.
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15.
16.
17.
18.
19.
20.
21.
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31.
32.
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35.
36.
37.
38.
39.
40.
41.
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43.
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46.
47.
48.
49.
50.
Chapter 18
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18. 1 = x
The solution of x = 1 must be rejected, so there is no solution.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.