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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. $$\frac{315}{120}$$

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. $$- ( - ( - ( - a ) ) )$$

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. $$x \leq - 1 \text { or } x > 3$$

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. $$( - \infty , 0 )$$

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. $$- ( - ( - | - 3 | ) )$$

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. $$| a | = a$$

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. $$\frac { - ( 5 - 2 | - 3 | ) ^ { 3 } } { \left| 4 - ( - 3 ) ^ { 2 } \right| - 3 ^ { 2 } }$$

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. $$\sqrt [ 3 ] { \frac { 1 } { 125 } }$$

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.

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. $$\left( 2 m ^ { 2 } - 3 m n + n ^ { 2 } \right) \cdot 6$$

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. $$5 m ^ { 2 } n - 3 m n + 2 m n ^ { 2 } - 2 n m - 4 m ^ { 2 } n + m n ^ { 2 }$$

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. $$m ^ { 2 } + m n - 6 \left( m ^ { 2 } - 3 n ^ { 2 } \right)$$

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.

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. $$\left( \frac { 6 a ^ { 3 } b ^ { - 3 } c } { 2 a ^ { 7 } b ^ { 0 } c ^ { - 4 } } \right) ^ { - 2 }$$

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.

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. $$\frac { 8 a ^ { 4 } - 10 } { a ^ { 2 } - 2 }$$

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?

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?

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.

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. $$\frac { 6 x ^ { 4 } - 17 x ^ { 3 } + 16 x ^ { 2 } - 18 x + 13 } { 2 x - 3 }$$

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 $$y : a x + b y = c$$

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. $$3 x - 7 > 14 \text { or } 3 x - 7 < - 14$$

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?
2. Length: $$31$$ inches; width: $$36$$ inches
4. The car can be driven at most $$125$$ miles.