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Mathematics LibreTexts

0.1e: Exercises - Real Number Operations

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    38206
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    A: The Number Line, Set and Interval Notation

    Exercises \(\PageIndex{1}\)

    \( \bigstar \) Graph the solution set and give the interval notation and set-builder notation equivalents.

    1.    \(x<−1\)

    2.    \(x>−3\)

    3.    \(x\geq −8\)

    4.    \(x\leq 6\)

    5.    \(−10\leq x<4\)

    6.    \(3<x\leq 7\)

    7.    \(−40<x<0\)

    8.    \(−12\leq x\leq −4\)

    9.    \(x<5\)  and  \(x\geq 0\)

    10.    \(x\leq −10\)  and  \(x\geq −40\)

    11.    \(x\leq 7\)  and  \(x<10\)

    12.    \(x<1\)  and  \(x>3\)

    13.    \(x<−2\)  or  \(x\geq 5\)

    14.    \(x\leq 0\)  or  \(x\geq 4\)

    15.    \(x<6\)  or  \(x>2\)

    16.    \(x<0\)  or  \(x\leq 5\)

    Answers to odd exercises:

    1. \((−∞, −1)\);   \(\{x| \, x<−1\}\);      

    e02b8833274c917789ccbb118b3d5eed.png
    Figure 0.1e.1

    3. \([8,∞)\);   \(\{x| \, x \geq 8\}\);

    f106d461744e41d767c257af071f92dd.png
    Figure 0.1e.3

    5. \([−10,4)\);  \(\{x|−10≤x<4\}\);

    85c6f49cb054143941dc82ec961b1e34.png
    Figure 0.1e.5

    7. \((−40,0)\);  \(\{x|−40<x<0\}\);

    00d4edd022e22e20603b2c8684e09806.png
    Figure 0.1e.7

    9. \([0,5)\);  \(\{x| \, 0≤x<5\}\);

    a634c6101a281506cf99247d9df48180.png
    Figure 0.1e.9

    11. \((−∞,7)\);   \(\{x| \, x<7\}\);

    f53685fcb168d52702d4f698cfdabc82.png
    Figure 0.1e.11

    13. \((−∞,−2)\cup [5,∞)\);    \(\{x| \, x<−2 \text{ or } x \geq 5\}\)

    c635102b40e626c903ba4be9aa3d81fa.png
    Figure 0.1e.13

    15. \((−∞,∞)=\mathbb{R}\);

    73cc7b4e93034a1e15a259f0810b6cea.png
    Figure 0.1e.15

    B: Convert a Description to an Inequality

    Exercises \(\PageIndex{2}\)

    \( \bigstar \)  Write an equivalent inequality.

    17.    All real numbers less than \(−15\).

    18.    All real numbers greater than or equal to \(−7\).

    19.    All real numbers less than \(6\) and greater than zero.

    20.    All real numbers less than zero and greater than \(−5\).

    21.    All real numbers less than or equal to \(5\) or greater than \(10\).

    22.    All real numbers between \(−2\) and \(2\).

    Answers to odd exercises:
    17. \(x<−15\) 19. \(0<x<6\) 21. \(x\leq 5\)  or  \(x>10\)

    C: Convert Interval Notation to an Inequality

    Exercises \(\PageIndex{3}\)

    \( \bigstar \) Determine the inequality given the answers expressed in interval notation.

    23.    \((−∞,12)\)

    24.    \([−8,∞)\)

    25.    \((−∞,0]\)

    26.    \((0,∞)\)

    27.    \((−6,14)\)

    28.    \((0,12]\)

    29.    \([5,25)\)

    30.    \([−30,−10]\)

    31.    \((−∞,2)\cup [3,∞)\)

    32.    \((−∞,−19]\cup [−12,∞)\)

    33.    \((−∞,−2)\cup (0,∞)\)

    34.    \((−∞,−15]\cup (−5,∞)\)

    Answers to odd exercises:
    23. \(x<12\)
    25. \(x\leq 0\)
    27. \(−6<x<14\)
    29. \(5\leq x<25\)
    31. \(x<2\)  or  \(x\geq 3\)
    33. \(x<−2\)  or  \(x>0\)

    D: Order of Operations

    Exercises \(\PageIndex{4}\)

    \( \bigstar \) Use order of operations to evaluate the given expression.

    36.    10+2×(53)

    37.    6÷2(81÷32)

    38.    18+(68)3

    39.    −2×[16÷(84)2]2

    40.    46+2×7

    41.    3(58)

    42.    4+610÷2

    43.    12÷(36÷9)+6

    44.    (4+5)2÷3

    45.    312×2+19

    46.    2+8×7÷4

    47.    5+(6+4)11

    48.    918÷32

    49.    14×3÷76

    50.    9(3+11)×2

    51.    6+2×21

    52.    64÷(8+4×2)

    53.    9+4(22)

    54.    (12÷3×3)2

    55.    25÷527

    56.    (157)×(37)

    57.    2×49(−1)

    58.    \(4^{2} - 25 \cdot \frac{1}{5}\)

    59.    12(31)÷6

    Answers to odd exercises:

    37.    6 \(\qquad\)39.    2 \(\qquad\)41.    9 \(\qquad\)43.    9 \(\qquad\)45.    −2 \(\qquad\)47.    4
    49.    0 \(\qquad\)51.    9 \(\qquad\)53.    25 \(\qquad\)55.    −6 \(\qquad\)57.    17 \(\qquad\)59.    4

     

    E: Substitute and Evaluate Expressions

    Exercises \(\PageIndex{5}\)

    \( \bigstar \) Evaluate.

    61.    \(−2x + 3\) where \(x = −2\)

    62.    \(8x − 5\) where \(x = −1\)

    63.    \(x^{2} − x + 5\) where \(x = −5\)

    64.    \(2x^{2} − 8x + 1\) where \(x = 3\)

    65.    \(\dfrac { x ^ { 2 } - x + 2 } { 2 x - 1 }\) where \(x = -\frac{1}{2}\)

    66.    \(\dfrac { 9 x ^ { 2 } + x - 2 } { 3 x - 4 }\) where \(x = -\frac{2}{3}\)

    67.    \(( 3 y - 2 ) ( y + 5 )\) where \(y = \frac { 2 } { 3 }\)

    68.    \((3x + 2) (5x + 1)\) where \(x = −\frac{1}{5}\)

    69.    \((3x − 1) (x − 8)\) where \(x = −1\)

    70.    \((7y + 5) (y + 1)\) where \(y = −2\)

    71.    \(y^{6} − y^{3} + 2\) where \(y = −1\)

    72.    \(y^{5} + y^{3} − 3\) where \(y = −2\)

    73.    \(a^{2} − 5b^{2}\) where \(a = −2\) and \(b = −1\)

    74.    \(a^{3} − 2b^{3}\) where \(a = −3\) and \(b = 2\)

    75.    \((x − 2y) (x + 2y)\) where \(x = 2\) and \(y = −5\)

    76.    \((4x − 3y) (x − y)\) where \(x = −4\) and \(y = −3\)

    77.    \(a^{2} − ab + b^{2}\) where \(a = −1\) and \(b = −2\)

    78.    \(x^{2}y^{2} − xy + 2\) where \(x = −3\) and \(y = −2\)

    79.    \(a^{4} − b^{4}\) where \(a = −2\) and \(b = −3\)

    80.    \(a^{6} − 2a^{3}b^{3} − b^{6}\) where \(a = 2\) and \(b = −1\)

    81. Evaluate \( \sqrt { b^2 - 4 a c }\) given the following values.

    a.    \(a = 6, b = 1\) and \(c = −1\)

    b.    \(a = 15, b = 4\) and \(c = −4\)

    c.    \(a = \dfrac{3}{4} , b = −2\) and \(c = −4\)

    d.    \(a = \dfrac{1}{2} , b = −2\) and \(c = −30\)

    e.    \(a = 1, b = 2\) and \(c = −1\)

    f.    \(a = 1, b = −4\) and \(c = −50\)

    g.    \(a = 1, b = −1\) and \(c = −\dfrac{1}{16}\)

    h.    \(a = −2, b = −\dfrac{1}{3}\) and \(c = 1\)

    Answers to odd exercises:

    61. \(7\) \(\qquad\) 63. \(35\) \(\qquad\) 65. \(−\frac{11}{8}\) \(\qquad\) 67. \(0\) \(\qquad\) 69. \(36\) \(\qquad\) 71. \(4\) \(\qquad\) 73. \(−1\) \(\qquad\) 
    75. \(−96\) \(\qquad\) 77. \(3\) \(\qquad\) 79. \(−65\) \(\qquad\) 81 a. \(5\) \(\qquad\) 81 c. \(4\) \(\qquad\) 81 e. \(2\sqrt{2}\) \(\qquad\) 81 g. \(\frac { \sqrt { 5 } } { 2 }\)

    F: Simplify Algebraic Expressions

    Exercises \(\PageIndex{6}\)

    \( \bigstar \) Simplify.

    85.    \(5 − 2 (4x + 8)\)

    86.    \(8 − 6 (2x − 1)\)

    87.    \(2 (x^{2} − 7x + 1) + 3x − 7\)

    88.    \(−5 (x^{2} + 4x − 1) + 8x^{2} − 5\)

    89.    \(5ab − 4 (ab + 5)\)

    90.    \(5 (7 − ab) + 2ab\)

    91.    \(2 − a^{2} + 3 (a^{2} + 4)\)

    92.    \(7 − 3y + 2 (y^{2} − 3y − 2)\)

    93.    \(8x^{2} − 3x − 5 (x^{2} + 4x − 1)\)

    94.    \(2 − 5y − 6 (y^{2} − y + 2)\)

    95.    \(a^{2}b^{2} − 5 + 3 (a^{2}b^{2} − 3ab + 2)\)

    96.    \(a^{2} − 3ab − 2 (a^{2} − ab + 1)\)

    97.    \(10y^{2} + 6 − (3y^{2} + 2y + 4)\)

    98.    \(4m^{2} − 3mn − (m^{2} − 3mn + n^{2} )\)

    99.    \(x^{2n} − 3x^{n} + 5 (x^{2n} − x^{n} + 1)\)

    100.    \(−3 (y^{2n} − 2y^{n} + 1) + 4y^{2n} − 5\)

    101.    \( 7x - x \div 2 \times 4  + \dfrac{x}{3} \)

    102.    \(5v \div 3v \times (9-6+2) \) 

    103.    \(7z - 3 + z \times 6^2 \)

    104.    \( 4 \times 3 + 18 x \div 9 - 12 \)

    105.    \( 8b -1 - 4b \times 3 + 2\)

    106.   \( a \div 64 \times 2^3  -12a \div 6\)

    107.    \(27 -(4)^2 y-11 \)

    108.    \(4x + x(13-7) \)

    109.    \( 9(y + 8) - 27 \) 

    110.    \( \Big{(} \displaystyle \frac{9}{6}t−4 \Big{)}2\)

    111.    \( 6 + 12b - 3 (6b) \)

    112.    \( 18y - 2(1 + 7y) \)

    113.   \( \Big{(} \displaystyle \frac{4}{9} \Big{)} ^{2} 27x \)

    114.    \( 8(3 - m) + 1(-8) \)

    115.    \( 9x + 4x(2 + 3) - 4(2x + 3x) \)

    116.    \( 5^2 - 4(3x) \)

    Answers to odd exercises:

    85. \(−8x − 11\) \( \qquad \) 87. \(2x^{2} − 11x − 5\) \( \qquad \) 89. \(ab − 20\) \( \qquad \) 91. \(2a^{2} + 14\)  \( \qquad \) 93. \(3x^{2} − 23x + 5\) \( \qquad \) 
    95. \(4a^{2}b^{2} − 9ab + 1\) \( \qquad \) 97. \(7y^{2} − 2y + 2\) \( \qquad \) 99. \(6x^{2n} − 8x^{n} + 5\) \( \qquad \)  101. \( \frac{16x}{3} \) \( \qquad\) 103. \(43z-3 \) 
    105. \(-4b+1\) \( \qquad \)  107. \( -14y-11\) \( \qquad \)  109. \(9y+45 \) 111. \( -6b+6 \)\( \qquad \)  113. \( \frac{16x}{3} \) \( \qquad \)  115. \(9x \)

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