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1.E: Algebra Fundamentals (Exercises)

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    Exercise \(\PageIndex{1}\)

    Reduce to lowest terms.

    1. \(\frac{56}{120}\)
    2. \(\frac{54}{60}\)
    3. \(\frac{155}{90}\)
    4. \(\frac{315}{120}\)
    Answer

    1. \(\frac{7}{15}\)

    3. \(\frac{31}{18}\)

    Exercise \(\PageIndex{2}\)

    Simplify.

    1. \(- \left( - \frac { 1 } { 2 } \right)\)
    2. \(- \left( - \left( - \frac { 5 } { 8 } \right) \right)\)
    3. \(- ( - ( - a ) )\)
    4. \(- ( - ( - ( - a ) ) )\)
    Answer

    1. \(\frac{1}{2}\)

    3. \(-a\)

    Exercise \(\PageIndex{3}\)

    Graph the solution set and give the interval notation equivalent.

    1. \(x \geq - 10\)
    2. \(x < 0\)
    3. \(- 8 \leq x < 0\)
    4. \(- 10 < x \leq 4\)
    5. \(x < 3 \text { and } x \geq - 1\)
    6. \(x < 0 \text { and } x > 1\)
    7. \(x < - 2 \text { or } x > - 6\)
    8. \(x \leq - 1 \text { or } x > 3\)
    Answer

    1. \([ - 10 , \infty )\);

    Figure 1.E.1

    3. \([ - 8,0 )\);

    Figure 1.E.2

    5. \([ - 1,3 )\);

    Figure 1.E.3

    7. \(\mathbb { R }\)

    Figure 1.E.4

    Exercise \(\PageIndex{4}\)

    Determine the inequality that corresponds to the set expressed using interval notation.

    1. \([ - 8 , \infty )\)
    2. \(( - \infty , - 7 )\)
    3. \([ 12,32 ]\)
    4. \([ - 10,0 )\)
    5. \(( - \infty , 1 ] \cup ( 5 , \infty )\)
    6. \(( - \infty , - 10 ) \cup ( - 5 , \infty )\)
    7. \(( - 4 , \infty )\)
    8. \(( - \infty , 0 )\)
    Answer

    1. \(x \geq - 8\)

    3. \(12 \leq x \leq 32\)

    5. \(x \leq 1 \text { or } x > 5\)

    7. \(x > - 4\)

    Exercise \(\PageIndex{5}\)

    Simplify.

    1. \(- \left| - \frac { 3 } { 4 } \right|\)
    2. \(- \left| - \left( - \frac { 2 } { 3 } \right) \right|\)
    3. \(- ( - | - 4 | )\)
    4. \(- ( - ( - | - 3 | ) )\)
    Answer

    1. \(-\frac{3}{4}\)

    3. \(4\)

    Exercise \(\PageIndex{6}\)

    Determine the values represented by \(a\).

    1. \(| a | = 6\)
    2. \(| a | = 1\)
    3. \(| a | = - 5\)
    4. \(| a | = a\)
    Answer

    1. \(a = \pm 6\)

    2. \(\varnothing\)

    Exercise \(\PageIndex{7}\)

    Perform the operations.

    1. \(\frac { 1 } { 4 } - \frac { 1 } { 5 } + \frac { 3 } { 20 }\)
    2. \(\frac { 2 } { 3 } - \left( - \frac { 3 } { 4 } \right) - \frac { 5 } { 12 }\)
    3. \(\frac { 5 } { 3 } \left( - \frac { 6 } { 7 } \right) \div \left( \frac { 5 } { 14 } \right)\)
    4. \(\left( - \frac { 8 } { 9 } \right) \div \frac { 16 } { 27 } \left( \frac { 2 } { 15 } \right)\)
    5. \(\left( - \frac { 2 } { 3 } \right) ^ { 3 }\)
    6. \(\left( - \frac { 3 } { 4 } \right) ^ { 2 }\)
    7. \(( - 7 ) ^ { 2 } - 8 ^ { 2 }\)
    8. \(- 4 ^ { 2 } + ( - 4 ) ^ { 3 }\)
    9. \(10 - 8 \left( ( 3 - 5 ) ^ { 2 } - 2 \right)\)
    10. \(4 + 5 \left( 3 - ( 2 - 3 ) ^ { 2 } \right)\)
    11. \(- 3 ^ { 2 } - \left( 7 - ( - 4 + 2 ) ^ { 3 } \right)\)
    12. \(( - 4 + 1 ) ^ { 2 } - ( 3 - 6 ) ^ { 3 }\)
    13. \(\frac { 10 - 3 ( - 2 ) ^ { 3 } } { 3 ^ { 2 } - ( - 4 ) ^ { 2 } }\)
    14. \(\frac { 6 \left[ ( - 5 ) ^ { 2 } - ( - 3 ) ^ { 2 } \right] } { 4 - 6 ( - 2 ) ^ { 2 } }\)
    15. \(7 - 3 \left| 6 - ( - 3 - 2 ) ^ { 2 } \right|\)
    16. \(- 6 ^ { 2 } + 5 \left| 3 - 2 ( - 2 ) ^ { 2 } \right|\)
    17. \(\frac { 12 - \left| 6 - 2 ( - 4 ) ^ { 2 } \right| } { 3 - | - 4 | }\)
    18. \(\frac { - ( 5 - 2 | - 3 | ) ^ { 3 } } { \left| 4 - ( - 3 ) ^ { 2 } \right| - 3 ^ { 2 } }\)
    Answer

    1. \(\frac{1}{5}\)

    3. \(-4\)

    5. \(-\frac{8}{27}\)

    7. \(-15\)

    9. \(-6\)

    11. \(-24\)

    13. \(-\frac{34}{7}\)

    15. \(-50\)

    17. \(14\)

    Exercise \(\PageIndex{8}\)

    Simplify.

    1. \(3 \sqrt { 8 }\)
    2. \(5 \sqrt { 18 }\)
    3. \(6 \sqrt { 0 }\)
    4. \(\sqrt { - 6 }\)
    5. \(\sqrt { \frac { 75 } { 16 } }\)
    6. \(\sqrt { \frac { 80 } { 49 } }\)
    7. \(\sqrt [ 3 ] { 40 }\)
    8. \(\sqrt [ 3 ] { 81 }\)
    9. \(\sqrt [ 3 ] { - 81 }\)
    10. \(\sqrt [ 3 ] { - 32 }\)
    11. \(\sqrt [ 3 ] { \frac { 250 } { 27 } }\)
    12. \(\sqrt [ 3 ] { \frac { 1 } { 125 } }\)
    Answer

    1. \(6 \sqrt { 2 }\)

    3. \(0\)

    5. \(\frac { 5 \sqrt { 3 } } { 4 }\)

    7. \(2 \sqrt [ 3 ] { 5 }\)

    9. \(- 3 \sqrt [ 3 ] { 3 }\)

    11. \(\frac { 5 \sqrt [ 3 ] { 2 } } { 3 }\)

    Exercise \(\PageIndex{9}\)

    Use a calculator to approximate the following to the nearest thousandth.

    1. \(\sqrt { 12 }\)
    2. \(3 \sqrt { 14 }\)
    3. \(\sqrt [ 3 ] { 18 }\)
    4. \(7 \sqrt [ 3 ] { 25 }\)
    5. Find the length of the diagonal of a square with sides measuring \(8\) centimeters.
    6. Find the length of the diagonal of a rectangle with sides measuring \(6\) centimeters and \(12\) centimeters.
    Answer

    1. \(3.464\)

    3. \(2.621\)

    5. \(8 \sqrt { 2 }\) centimeters

    Exercise \(\PageIndex{10}\)

    Multiply

    1. \(\frac { 2 } { 3 } \left( 9 x ^ { 2 } + 3 x - 6 \right)\)
    2. \(- 5 \left( \frac { 1 } { 5 } y ^ { 2 } - \frac { 3 } { 5 } y + \frac { 1 } { 2 } \right)\)
    3. \(\left( a ^ { 2 } - 5 a b - 2 b ^ { 2 } \right) ( - 3 )\)
    4. \(\left( 2 m ^ { 2 } - 3 m n + n ^ { 2 } \right) \cdot 6\)
    Answer

    1. \(6 x ^ { 2 } + 2 x - 4\)

    3. \(- 3 a ^ { 2 } + 15 a b + 6 b ^ { 2 }\)

    Exercise \(\PageIndex{11}\)

    Combine like terms.

    1. \(5 x ^ { 2 } y - 3 x y ^ { 2 } - 4 x ^ { 2 } y - 7 x y ^ { 2 }\)
    2. \(9 x ^ { 2 } y ^ { 2 } + 8 x y + 3 - 5 x ^ { 2 } y ^ { 2 } - 8 x y - 2\)
    3. \(a ^ { 2 } b ^ { 2 } - 7 a b + 6 - a ^ { 2 } b ^ { 2 } + 12 a b - 5\)
    4. \(5 m ^ { 2 } n - 3 m n + 2 m n ^ { 2 } - 2 n m - 4 m ^ { 2 } n + m n ^ { 2 }\)
    Answer

    1. \(x ^ { 2 } y - 10 x y ^ { 2 }\)

    3. \(5 a b + 1\)

    Exercise \(\PageIndex{12}\)

    Simplify.

    1. \(5 x ^ { 2 } + 4 x - 3 \left( 2 x ^ { 2 } - 4 x - 1 \right)\)
    2. \(\left( 6 x ^ { 2 } y ^ { 2 } + 3 x y - 1 \right) - \left( 7 x ^ { 2 } y ^ { 2 } - 3 x y + 2 \right)\)
    3. \(a ^ { 2 } - b ^ { 2 } - \left( 2 a ^ { 2 } + a b - 3 b ^ { 2 } \right)\)
    4. \(m ^ { 2 } + m n - 6 \left( m ^ { 2 } - 3 n ^ { 2 } \right)\)
    Answer

    1. \(- x ^ { 2 } + 16 x + 3\)

    3. \(- a ^ { 2 } - a b + 2 b ^ { 2 }\)

    Exercise \(\PageIndex{13}\)

    Evaluate.

    1. \(x ^ { 2 } - 3 x + 1 \text { where } x = - \frac { 1 } { 2 }\)
    2. \(x ^ { 2 } - x - 1 \text { where } x = - \frac { 2 } { 3 }\)
    3. \(a ^ { 4 } - b ^ { 4 } \text { where } a = - 3 \text { and } b = - 1\)
    4. \(a ^ { 2 } - 3 a b + 5 b ^ { 2 } \text { where } a = 4 \text { and } b = - 2\)
    5. \(( 2 x + 1 ) ( x - 3 ) \text { where } x = - 3\)
    6. \(( 3 x + 1 ) ( x + 5 ) \text { where } x = - 5\)
    7. \(\sqrt { b ^ { 2 } - 4 a c } \text { where } a = 2 , b = - 4 , \text { and } c = - 1\)
    8. \(\sqrt { b ^ { 2 } - 4 a c } \text { where } a = 3 , b = - 6 , \text { and } c = - 2\)
    9. \(\pi r ^ { 2 } h \text { where } r = 2 \sqrt { 3 } \text { and } h = 5\)
    10. \(\frac { 4 } { 3 } \pi r ^ { 3 } \text { where } r = 2 \sqrt [ 3 ] { 6 }\)
    11. What is the simple interest earned on a \(4\) year investment of \($4,500\) at an annual interest rate of \(4 \frac{3}{4}\)%?
    12. James traveled at an average speed of \(48\) miles per hour for \(2 \frac{1}{4}\) hours. How far did he travel?
    13. The period of a pendulum \(T\) in seconds is given by the formula \(T = 2 \pi \sqrt { \frac { L } { 32 } }\) where \(L\) represents its length in feet. Approximate the period of a pendulum with length \(2\) feet. Round off to the nearest tenth of a foot.
    14. The average distance \(d\), in miles, a person can see an object is given by the formula \(d = \frac { \sqrt { 6 h } } { 2 }\) where \(h\) represents the person’s height above the ground, measured in feet. What average distance can a person see an object from a height of \(10\) feet? Round off to the nearest tenth of a mile.
    Answer

    1. \(\frac{11}{4}\)

    3. \(80\)

    5. \(30\)

    7. \(2 \sqrt { 6 }\)

    9. \(60 \pi\)

    11. \(\$ 855\)

    13. \(1.6\) seconds

    Exercise \(\PageIndex{14}\)

    Multiply.

    1. \(\frac { x ^ { 10 } \cdot x ^ { 2 } } { x ^ { 5 } }\)
    2. \(\frac { x ^ { 6 } \left( x ^ { 2 } \right) ^ { 4 } } { x ^ { 3 } }\)
    3. \(- 7 x ^ { 2 } y z ^ { 3 } \cdot 3 x ^ { 4 } y ^ { 2 } z\)
    4. \(3 a ^ { 2 } b ^ { 3 } c \left( - 4 a ^ { 2 } b c ^ { 4 } \right) ^ { 2 }\)
    5. \(\frac { - 10 a ^ { 5 } b ^ { 0 } c ^ { - 4 } } { 25 a ^ { - 2 } b ^ { 2 } c ^ { - 3 } }\)
    6. \(\frac { - 12 x ^ { - 6 } y ^ { - 2 } z } { 36 x ^ { - 3 } y ^ { 4 } z ^ { 6 } }\)
    7. \(\left( - 2 x ^ { - 5 } y ^ { - 3 } z \right) ^ { - 4 }\)
    8. \(\left( 3 x ^ { 6 } y ^ { - 3 } z ^ { 0 } \right) ^ { - 3 }\)
    9. \(\left( \frac { - 5 a ^ { 2 } b ^ { 3 } } { c ^ { 5 } } \right) ^ { 2 }\)
    10. \(\left( \frac { - 3 m ^ { 5 } } { 5 n ^ { 2 } } \right) ^ { 3 }\)
    11. \(\left( \frac { - 2 a ^ { - 2 } b ^ { 3 } c } { 3 a b ^ { - 2 } c ^ { 0 } } \right) ^ { - 3 }\)
    12. \(\left( \frac { 6 a ^ { 3 } b ^ { - 3 } c } { 2 a ^ { 7 } b ^ { 0 } c ^ { - 4 } } \right) ^ { - 2 }\)
    Answer

    1. \(x ^ { 7 }\)

    3. \(- 21 x ^ { 6 } y ^ { 3 } z ^ { 4 }\)

    5. \(- \frac { 2 a ^ { 7 } } { 5 b ^ { 2 } c }\)

    7. \(\frac { x ^ { 20 } y ^ { 12 } } { 16 z ^ { 4 } }\)

    9. \(\frac { 25 a ^ { 4 } b ^ { 6 } } { c ^ { 10 } }\)

    11. \(- \frac { 27 a ^ { 9 } } { 8 b ^ { 15 } c ^ { 3 } }\)

    Exercise \(\PageIndex{15}\)

    Perform the operations.

    1. \(\left( 4.3 \times 10 ^ { 22 } \right) \left( 3.1 \times 10 ^ { - 8 } \right)\)
    2. \(\left( 6.8 \times 10 ^ { - 33 } \right) \left( 1.6 \times 10 ^ { 7 } \right)\)
    3. \(\frac { 1.4 \times 10 ^ { - 32 } } { 2 \times 10 ^ { - 10 } }\)
    4. \(\frac { 1.15 \times 10 ^ { 26 } } { 2.3 \times 10 ^ { - 7 } }\)
    5. The value of a new tablet computer in dollars can be estimated using the formula \(v = 450(t + 1)^{ −1}\) where \(t\) represents the number of years after it is purchased. Use the formula to estimate the value of the tablet computer \(2 \frac{1}{2}\) years after it was purchased.
    6. The speed of light is approximately \(6.7 × 10^{8}\) miles per hour. Express this speed in miles per minute and determine the distance light travels in \(4\) minutes.
    Answer

    1. \(1.333 \times 10 ^ { 15 }\)

    3. \(7 \times 10 ^ { - 23 }\)

    5. \(\$ 128.57\)

    Exercise \(\PageIndex{16}\)

    Simplify.

    1. \(\left( x ^ { 2 } + 3 x - 5 \right) - \left( 2 x ^ { 2 } + 5 x - 7 \right)\)
    2. \(\left( 6 x ^ { 2 } - 3 x + 5 \right) + \left( 9 x ^ { 2 } + 3 x - 4 \right)\)
    3. \(\left( a ^ { 2 } b ^ { 2 } - a b + 6 \right) - ( a b + 9 ) + \left( a ^ { 2 } b ^ { 2 } - 10 \right)\)
    4. \(\left( x ^ { 2 } - 2 y ^ { 2 } \right) - \left( x ^ { 2 } + 3 x y - y ^ { 2 } \right) - \left( 3 x y + y ^ { 2 } \right)\)
    5. \(- \frac { 3 } { 4 } \left( 16 x ^ { 2 } + 8 x - 4 \right)\)
    6. \(6 \left( \frac { 4 } { 3 } x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 5 } { 6 } \right)\)
    7. \(( 2 x + 5 ) ( x - 4 )\)
    8. \(( 3 x - 2 ) \left( x ^ { 2 } - 5 x + 2 \right)\)
    9. \(\left( x ^ { 2 } - 2 x + 5 \right) \left( 2 x ^ { 2 } - x + 4 \right)\)
    10. \(\left( a ^ { 2 } + b ^ { 2 } \right) \left( a ^ { 2 } - b ^ { 2 } \right)\)
    11. \(( 2 a + b ) \left( 4 a ^ { 2 } - 2 a b + b ^ { 2 } \right)\)
    12. \(( 2 x - 3 ) ^ { 2 }\)
    13. \(( 3 x - 1 ) ^ { 3 }\)
    14. \(( 2 x + 3 ) ^ { 4 }\)
    15. \(\left( x ^ { 2 } - y ^ { 2 } \right) ^ { 2 }\)
    16. \(\left( x ^ { 2 } y ^ { 2 } + 1 \right) ^ { 2 }\)
    17. \(\frac { 27 a ^ { 2 } b - 9 a b + 81 a b ^ { 2 } } { 3 a b }\)
    18. \(\frac { 125 x ^ { 3 } y ^ { 3 } - 25 x ^ { 2 } y ^ { 2 } + 5 x y ^ { 2 } } { 5 x y ^ { 2 } }\)
    19. \(\frac { 2 x ^ { 3 } - 7 x ^ { 2 } + 7 x - 2 } { 2 x - 1 }\)
    20. \(\frac { 12 x ^ { 3 } + 5 x ^ { 2 } - 7 x - 3 } { 4 x + 3 }\)
    21. \(\frac { 5 x ^ { 3 } - 21 x ^ { 2 } + 6 x - 3 } { x - 4 }\)
    22. \(\frac { x ^ { 4 } + x ^ { 3 } - 3 x ^ { 2 } + 10 x - 1 } { x + 3 }\)
    23. \(\frac { a ^ { 4 } - a ^ { 3 } + 4 a ^ { 2 } - 2 a + 4 } { a ^ { 2 } + 2 }\)
    24. \(\frac { 8 a ^ { 4 } - 10 } { a ^ { 2 } - 2 }\)
    Answer

    1. \(- x ^ { 2 } - 2 x + 2\)

    3. \(2 a ^ { 2 } b ^ { 2 } - 2 a b - 13\)

    5. \(- 12 x ^ { 2 } - 6 x + 3\)

    7. \(2 x ^ { 2 } - 3 x - 20\)

    9. \(2 x ^ { 4 } - 5 x ^ { 3 } + 16 x ^ { 2 } - 13 x + 20\)

    11. \(8 a ^ { 3 } + b ^ { 3 }\)

    13. \(27 x ^ { 3 } - 27 x ^ { 2 } + 9 x - 1\)

    15. \(x ^ { 4 } - 2 x ^ { 2 } y ^ { 2 } + y ^ { 4 }\)

    17. \(9 a + 27 b - 3\)

    19. \(x ^ { 2 } - 3 x + 2\)

    21. \(5 x ^ { 2 } - x + 2 + \frac { 5 } { x - 4 }\)

    23. \(a ^ { 2 } - a + 2\)

    Exercise \(\PageIndex{17}\)

    Solve.

    1. \(6 x - 8 = 2\)
    2. \(12 x - 5 = 3\)
    3. \(\frac { 5 } { 4 } x - 3 = \frac { 1 } { 2 }\)
    4. \(\frac { 5 } { 6 } x - \frac { 1 } { 4 } = \frac { 3 } { 2 }\)
    5. \(\frac { 9 x + 2 } { 3 } = \frac { 5 } { 6 }\)
    6. \(\frac { 3 x - 8 } { 10 } = \frac { 5 } { 2 }\)
    7. \(3 a - 5 - 2 a = 4 a - 6\)
    8. \(8 - 5 y + 2 = 4 - 7 y\)
    9. \(5 x - 6 - 8 x = 1 - 3 x\)
    10. \(17 - 6 x - 10 = 5 x + 7 - 11 x\)
    11. \(5 ( 3 x + 3 ) - ( 10 x - 4 ) = 4\)
    12. \(6 - 2 ( 3 x - 1 ) = - 4 ( 1 - 3 x )\)
    13. \(9 - 3 ( 2 x + 3 ) + 6 x = 0\)
    14. \(- 5 ( x + 2 ) - ( 4 - 5 x ) = 1\)
    15. \(\frac { 5 } { 9 } ( 6 y + 27 ) = 2 - \frac { 1 } { 3 } ( 2 y + 3 )\)
    16. \(4 - \frac { 4 } { 5 } ( 3 a + 10 ) = \frac { 1 } { 10 } ( 4 - 2 a )\)
    17. Solve for \(s : A = \pi r ^ { 2 } + \pi r s\)
    18. Solve for \(x : y = m x + b\)
    19. A larger integer is \(3\) more than twice another. If their sum divided by \(2\) is \(9\), find the integers.
    20. The sum of three consecutive odd integers is \(171\). Find the integers.
    21. The length of a rectangle is \(3\) meters less than twice its width. If the perimeter measures \(66\) meters, find the length and width.
    22. How long will it take \($500\) to earn \($124\) in simple interest earning \(6.2\)% annual interest?
    23. It took Sally \(3 \frac{1}{2}\) hours to drive the \(147\) miles home from her grandmother’s house. What was her average speed?
    24. Jeannine invested her bonus of \($8,300\) in two accounts. One account earned \(3 \frac{1}{2}\) % simple interest and the other earned \(4 \frac{3}{4}\) % simple interest. If her total interest for one year was \($341.75\), how much did she invest in each account?
    Answer

    1. \(\frac{5}{3}\)

    3. \(\frac{14}{5}\)

    5. \(\frac{1}{18}\)

    7. \(\frac{1}{3}\)

    9. \(\varnothing\)

    11. \(-3\)

    13. \(\mathbb { R }\)

    15. \(-\frac{7}{2}\)

    17. \(s = \frac { A - \pi r ^ { 2 } } { \pi r }\)

    19. \(5,13\)

    21. Length: \(21\) meters; Width: \(12\) meters

    23. \(42\) miles per hour

    Exercise \(\PageIndex{18}\)

    Solve. Graph all solutions on a number line and provide the corresponding interval notation.

    1. \(5 x - 7 < 18\)
    2. \(2 x - 1 > 2\)
    3. \(9 - x \leq 3\)
    4. \(3 - 7 x \geq 10\)
    5. \(61 - 3 ( x + 3 ) > 13\)
    6. \(7 - 3 ( 2 x - 1 ) \geq 6\)
    7. \(\frac { 1 } { 3 } ( 9 x + 15 ) - \frac { 1 } { 2 } ( 6 x - 1 ) < 0\)
    8. \(\frac { 2 } { 3 } ( 12 x - 1 ) + \frac { 1 } { 4 } ( 1 - 32 x ) < 0\)
    9. \(20 + 4 ( 2 a - 3 ) \geq \frac { 1 } { 2 } a + 2\)
    10. \(\frac { 1 } { 3 } \left( 2 x + \frac { 3 } { 2 } \right) - \frac { 1 } { 4 } x < \frac { 1 } { 2 } \left( 1 - \frac { 1 } { 2 } x \right)\)
    11. \(- 4 \leq 3 x + 5 < 11\)
    12. \(5 < 2 x + 15 \leq 13\)
    13. \(- 1 < 4 ( x + 1 ) - 1 < 9\)
    14. \(0 \leq 3 ( 2 x - 3 ) + 1 \leq 10\)
    15. \(- 1 < \frac { 2 x - 5 } { 4 } < 1\)
    16. \(- 2 \leq \frac { 3 - x } { 3 } < 1\)
    17. \(2 x + 3 < 13 \text { and } 4 x - 1 > 10\)
    18. \(3 x - 1 \leq 8 \text { and } 2 x + 5 \geq 23\)
    19. \(5 x - 3 < - 2 \text { or } 5 x - 3 > 2\)
    20. \(1 - 3 x \leq - 1 \text { or } 1 - 3 x \geq 1\)
    21. \(5 x + 6 < 6 \text { or } 9 x - 2 > - 11\)
    22. \(2 ( 3 x - 1 ) < - 16 \text { or } 3 ( 1 - 2 x ) < - 15\)
    23. Jerry scored \(90, 85, 92\), and \(76\) on the first four algebra exams. What must he score on the fifth exam so that his average is at least \(80\)?
    24. If \(6\) degrees less than \(3\) times an angle is between \(90\) degrees and \(180\) degrees, then what are the bounds of the original angle?
    Answer

    1. \(( - \infty , 5 )\);

    Figure 1.E.5

    3. \([ 6 , \infty )\);

    Figure 1.E.6

    5. \(( - \infty , 13 )\);

    Figure 1.E.7

    7. \(\varnothing\);

    Figure 1.E.8

    9. \(\left[ - \frac { 4 } { 5 } , \infty \right)\);

    Figure 1.E.9

    11. \([ - 3,2 )\);

    Figure 1.E.10

    13. \(\left( - 1 , \frac { 3 } { 2 } \right)\);

    Figure 1.E.11

    15. \(\left( \frac { 1 } { 2 } , \frac { 9 } { 2 } \right)\);

    Figure 1.E.12

    17. \(\left( \frac { 11 } { 4 } , 5 \right)\);

    Figure 1.E.13

    19. \(\left( - \infty , \frac { 1 } { 5 } \right) \cup ( 1 , \infty )\);

    Figure 1.E.14

    21. \(\mathbb { R }\);

    Figure 1.E.15

    23. Jerry must score at least \(57\) on the fifth exam.

    Sample Exam

    Exercise \(\PageIndex{19}\)

    Simplify.

    1. \(5 - 3 \left( 12 - \left| 2 - 5 ^ { 2 } \right| \right)\)
    2. \(\left( - \frac { 1 } { 2 } \right) ^ { 2 } - \left( 3 - 2 \left| - \frac { 3 } { 4 } \right| \right) ^ { 3 }\)
    3. \(- 7 \sqrt { 60 }\)
    4. \(5 \sqrt [ 3 ] { - 32 }\)
    5. Find the diagonal of a square with sides measuring \(6\) centimeters.
    Answer

    1. \(38\)

    3. \(- 14 \sqrt { 15 }\)

    5. \(6 \sqrt { 2 }\) centimeters

    Exercise \(\PageIndex{20}\)

    Simplify

    1. \(- 5 x ^ { 2 } y z ^ { - 1 } \left( 3 x ^ { 3 } y ^ { - 2 } z \right)\)
    2. \(\left( \frac { - 2 a ^ { - 4 } b ^ { 2 } c } { a ^ { - 3 } b ^ { 0 } c ^ { 2 } } \right) ^ { - 3 }\)
    3. \(2 \left( 3 a ^ { 2 } b ^ { 2 } + 2 a b - 1 \right) - a ^ { 2 } b ^ { 2 } + 2 a b - 1\)
    4. \(\left( x ^ { 2 } - 6 x + 9 \right) - \left( 3 x ^ { 2 } - 7 x + 2 \right)\)
    5. \(( 2 x - 3 ) ^ { 3 }\)
    6. \(( 3 a - b ) \left( 9 a ^ { 2 } + 3 a b + b ^ { 2 } \right)\)
    7. \(\frac { 6 x ^ { 4 } - 17 x ^ { 3 } + 16 x ^ { 2 } - 18 x + 13 } { 2 x - 3 }\)
    Answer

    2. \(- \frac { a ^ { 3 } c ^ { 3 } } { 8 b ^ { 6 } }\)

    4. \(- 2 x ^ { 2 } + x + 7\)

    6. \(27 a ^ { 3 } - b ^ { 3 }\)

    Exercise \(\PageIndex{21}\)

    Solve.

    1. \(\frac { 4 } { 5 } x - \frac { 2 } { 15 } = 2\)
    2. \(\frac { 3 } { 4 } ( 8 x - 12 ) - \frac { 1 } { 2 } ( 2 x - 10 ) = 16\)
    3. \(12 - 5 ( 3 x - 1 ) = 2 ( 4 x + 3 )\)
    4. \(\frac { 1 } { 2 } ( 12 x - 2 ) + 5 = 4 \left( \frac { 3 } { 2 } x - 8 \right)\)
    5. Solve for \(y : a x + b y = c\)
    Answer

    1. \(\frac{8}{3}\)

    3. \(\frac{11}{23}\)

    5. \(y = \frac { c - a x } { b }\)

    Exercise \(\PageIndex{22}\)

    Solve. Graph the solutions on a number line and give the corresponding interval notation.

    1. \(2 ( 3 x - 5 ) - ( 7 x - 3 ) \geq 0\)
    2. \(2 ( 4 x - 1 ) - 4 ( 5 + 2 x ) < - 10\)
    3. \(- 6 \leq \frac { 1 } { 4 } ( 2 x - 8 ) < 4\)
    4. \(3 x - 7 > 14 \text { or } 3 x - 7 < - 14\)
    Answer

    2. \(\mathbb { R }\);

    Figure 1.E.16

    4. \(\left( - \infty , - \frac { 7 } { 3 } \right) \cup ( 7 , \infty )\);

    Figure 1.E.17

    Exercise \(\PageIndex{23}\)

    Use algebra to solve the following.

    1. Degrees Fahrenheit \(F\) is given by the formula \(F = \frac{9}{5} C + 32\) where C represents degrees Celsius. What is the Fahrenheit equivalent to \(35\)° Celsius?
    2. The length of a rectangle is \(5\) inches less than its width. If the perimeter is \(134\) inches, find the length and width of the rectangle.
    3. Melanie invested \(4,500\) in two separate accounts. She invested part in a CD that earned \(3.2\)% simple interest and the rest in a savings account that earned \(2.8\)% simple interest. If the total simple interest for one year was \($138.80\), how much did she invest in each account?
    4. A rental car costs \($45.00\) per day plus \($0.48\) per mile driven. If the total cost of a one-day rental is to be at most \($105\), how many miles can be driven?
    Answer

    2. Length: \(31\) inches; width: \(36\) inches

    4. The car can be driven at most \(125\) miles.

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