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Mathematics LibreTexts

1.5: Exercises

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Exercise 1.5.1

Give examples of numbers that are

  1. natural numbers
  2. integers
  3. integers but not natural numbers
  4. rational numbers
  5. real numbers
  6. rational numbers but not integers
Answer
  1. 2,3,5
  2. 3,0,6
  3. 3,4,0
  4. 23,47,8
  5. 5,π,331
  6. 12,25,0.75

Exercise 1.5.2

Which of the following numbers are natural numbers, integers, rational numbers, or real numbers? Which of these numbers are irrational?

  1. 73
  2. 5
  3. 0
  4. 17,000
  5. 124
  6. 7
  7. 25
Answer
  1. rational
  2. integer, rational
  3. integer, rational
  4. natural, integer, rational
  5. natural, integer, rational
  6. irrational
  7. natural, integer, rational

All of the given numbers are real numbers

Exercise 1.5.3

Evaluate the following absolute value expressions:

  1. |8|
  2. |10|
  3. |99|
  4. |3|
  5. |2|
  6. |6|
  7. |3+4|
  8. |29|
  9. |5.4|
  10. |23|
  11. |52|
  12. |63|
Answer
  1. 8
  2. 10
  3. 99
  4. 3
  5. 2
  6. 6
  7. 7
  8. 7
  9. 5.4
  10. 23
  11. 52
  12. 2

Exercise 1.5.4

Solve for x:

  1. |x|=8
  2. |x|=0
  3. |x|=3
  4. |x+3|=10
  5. |2x+5|=9
  6. |25x|=22
  7. |4x|=8
  8. |7x3|=0
  9. |44x|=44
  10. 2|23x|=12
  11. 5+|2x+7|=14
  12. |82x|=12
Answer
  1. S={8,8}
  2. S={0}
  3. S={}
  4. S={13,7}
  5. S={7,2}
  6. S={4,245}
  7. S={}
  8. S={37}
  9. S={10,12}
  10. S={43,83}
  11. S={8,1}
  12. S={10,2}

Exercise 1.5.5

Solve for x using the geometric interpretation of the absolute value:

  1. |x|=8
  2. |x|=0
  3. |x|=3
  4. |x4|=2
  5. |x+5|=9
  6. |2x|=5
Answer
  1. S={8,8}
  2. S={0}
  3. S={}
  4. S={2,6}
  5. S={14,4}
  6. S={3,7}

Exercise 1.5.6

Complete the table.

Inequality notation Number line Interval notation
2x<5    
x3    
  clipboard_ee36fbb8a1140f74a4f63d1e8e3ff679a.png  
  clipboard_e6c534d4d0379a6b2fddb24f85dc1f57e.png  
    [2,6]
    (,0)
  clipboard_ee530401a798598d9e9a7c8efd7988e37.png  
5<x30    
    (137,π)
Answer
Inequality notation Number line Interval notation
2x<5 clipboard_ecc556141ef9c6812ed11f16885af51e9.png [2,5)
x3 clipboard_e7d2ae6b9269b96fcb9abdf5ab7067ab5.png (,3]
12<x17 clipboard_ee36fbb8a1140f74a4f63d1e8e3ff679a.png (12,17]
x<2 clipboard_e6c534d4d0379a6b2fddb24f85dc1f57e.png (,2)
2x6 clipboard_e98502b65d85cdc5539d27cf17265fc06.png [2,6]
x<0 clipboard_ecc1e5c8860f45a8764dba047a1cdfcb3.png (,0)
4.5x clipboard_ee530401a798598d9e9a7c8efd7988e37.png [4.5,)
5<x30 clipboard_ea672b8a6eaaa02d6a68eec34566831e7.png (5,30]
137<x<π clipboard_edc9edf9b56a08bf45919a6fc47142c9a.png (137,π)

Exercise 1.5.7

Solve for x and write the solution in interval notation.

  1. |x4|7
  2. |x4|7
  3. |x4|>7
  4. |2x+7|13
  5. |24x|>8
  6. |4x+2|<17
  7. |153x|6
  8. |5x43|>23
  9. |2x2|8
  10. |2x+3|<5
  11. |5+5x|2
  12. |5+5x|>0
Answer
  1. S=[3,11]
  2. S=(,3][11,)
  3. S=(,3)(11,)
  4. S=[10,3],
  5. S=(,52)(32,)
  6. S=(194,154)
  7. S=(,3][7,)
  8. S=(,215)(25,)
  9. S=[1,3]
  10. S={}
  11. S=(,)=R
  12. S=(,1)(1,)=R{1}

This page titled 1.5: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.

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