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MATH 081

REVIEW MATERIALS

YOU WILL NOT BE GIVEN A TEST UNTIL THIS MATERIAL IS RETURNED.

DO NOT WRITE IN THIS MATERIAL. Math 081 Review Materials 2

Revised 6/00 MATH 081 PRACTICE TEST

CALCULATORS ARE NOT PERMITTED ON THIS TEST. ANSWERS

1. 963 2. 721 3. 453 1. 1401 + 438 - 368 x 84 2. 353 3. 38,052 4. 72  9 = 4. 8 p. 5

5. 7 2135 5. 305 p. 5

6. What is the quotient and remainder for 23 47175 ? 6. 2051 R2 p. 5

7. A teacher gives 15 children some candy. If there are 75 pieces 7. 5 p. 5 of candy, how many pieces does each child get?

8. Are the fractions and equal ? Show why or why not. 8. yes

9. Simplify: 9.

10. Write as a mixed numeral. 10. 5

11. Write 4 as an improper fraction. 11.

12. Which fraction is smaller, or ? 12. p. 8

13. + = 13. p. 9

14. + = 14. p. 9

15. 5 15. 8 p. 9 + 2 Math 081 Review Materials 3

Math 081 Practice Test cont.

16. 4 16. 1 p. 10 - 2

17. A person cuts 1 feet from a 6-foot board. How much is left? 17. 4 feet p. 10

18. x = 18. p. 11

19. x 2 = 19. or 1 p. 11

20. John earns of what Harry makes. If Harry made $120 20. $80 p. 11 one week, how much did John make?

21.  21. p. 11 22.  3 22. p. 11

23. Sue has 10 pounds of flour that she put into pound 23. 20 p. 11 containers. How many containers did she fill?

24. In the number 526.103, which digit is in the tenths place? 24. 1 p. 12

25. Fill in the correct symbol: 4.74 _____ 4.736 25. > p. 12 > is “greater than”, < is “less than”, = is “equal to”

26. 1 + .2 + .45 26. 1.65 p. 13

27. 4 – 1.56 27. 2.44 p. 13

28. Round 4.683 to the tenths place. 28. 4.7 p. 14

29. 1.45 29. 3.335 p. 14 x 2 .3

30. 1.05 37.38 (Divide until there is no remainder.) 30. 35.6 p. 15

Math 081 Practice Test cont.

31. Find the highest common factor of 8 and 12. 31. 4 p. 16 32. Find the least common multiple of 8 and 12. 32. 24 p. 16 33. Evaluate (find the result): -8 + 5 33. -3 p. 17 34. What is the opposite of 5? 34. -5 p. 18 Math 081 Review Materials 4

35. Evaluate: (-8) – (-3) 35. -5 p. 18

36. Evaluate: (-4)  (6) 36. -24 p. 19

37. Evaluate: 37. 4 p. 20

38. Evaluate: 1 + 5  2 38. 11 p. 20 39. Evaluate: 3 + [ 2 (9 – 1) + 1 ] 39. 20 p. 20 40. Evaluate: 32 40. 9 p. 21 41. Evaluate: -32 41. -9 p. 21 42. Evaluate: (-3)2 42. 9 p. 21 43. Evaluate: x(x + 3) when x = 2 43. 10 p. 22 44. Evaluate: 5x – 2y when x = 1 and y = 2 44. 1 p. 22 45. Evaluate: 2x + 1 when x = -2 45. -3 p. 22 46. Simplify: -x – 5 + 3x + 2 46. 2x – 3 p. 23 47. Simplify: 2 (3x – 4) – 5 (x + 1) 47. x – 13 p. 23 48. Simplify: x  x2  x3 48. x6 p. 24

49. Solve for x: x – 1 = 2 49. 3 p. 25 50. Solve for x: -2x + 3 = 5 50. -1 p. 25 51. Solve for x: 3x – 1 = 5x – 7 51. 3 p. 25 52. Solve for x: 3(2x – 1) = -15 – (x + 2) 52. -2 p. 25 Math 081 Review Materials 5

I. DIVISION OF WHOLE NUMBERS

A. Examples:

1. Note the steps involved in dividing 2303 by 8:

287 28 2 8 2303 8 2303 16 8 2303 16 70 16 70 64 70 64 63 63 56 7 2303 divided by 8 = 287 remainder 7 To check, multiply 287 by 8 and then add 7. You will get 2303.

30 12 360 2. 36 0 0

Thus 360  12 = 30 (no remainder) To check, multiply 12 by 30. You will get 360.

805 51 41058 408 3. 25 0 258 255 3 To check, multiply 51 by 805 and then add 3. You will get 41058.

B. Exercises:

1. divide 52 by 3 1. 17 R1

2. 4 420 2. 105

3. 12 827 3. 68 R11 4. 225  5 = 4. 45

5. 18 3730 5. 207 R4 6. 1155  23 = 6. 50 R5 Math 081 Review Materials 6

II. MEANING OF FRACTIONS

A. Examples:

1. What fractional portion of this figure is shaded?

Shaded parts: 1 part Total parts: 2 parts

is the fractional portion which is shaded.

1 is the numerator and 2 is the denominator.

B. Exercises: What fractional part is shaded for each of these? 1. 2.

III. Equivalent Fractions

A. Examples: 1. To obtain fractions equivalent to , multiply the numerator and denominator by the same number.

= ; = ; = ; =

Thus, = = = = and so forth.

2. is equivalent to since

6  12 = 72 8  9 = 72

Since 6  12 = 8  9, the fractions are equivalent.

3. is not equivalent to since

2  15 = 30 3  9 = 27

Since 2  15 is not equal to 3  9, the fractions are not equivalent.

B. Exercises: 1. Name the first five fractions which are equivalent to 1. , , , , Math 081 Review Materials 7

2. Determine whether the following pairs of fractions are equivalent or not. a. and 2a. Yes, since 6  15 = 9  10

b. and b. No, since 15  30 = 450 and 20  18 = 360

IV. Simplifying Fractions

A. Examples: 1. is in simplest form, since 18 and 25 have no common divisor (other than 1).

2. is not in simplest form, since 5 is a divisor of both 10 and 25. Divide numerator and denominator by 5 to simplify. =

B. Exercises: Write the following fractions in simplest form.

V. Mixed Numerals and Improper Fractions

A. Examples:

1. Express the improper fraction as a mixed numeral.

7 Quotient Divisor 5 38 thus = 7 Remainder 35 Divisor 3 Remainder

Quotient

2. Express the improper fraction as a mixed numeral. 87 2 175 16 thus = 87 15 14 1

3. Express the mixed numeral 2 as an improper fraction. Math 081 Review Materials 8

2 = = =

4. Express the mixed numeral 7 as an improper fraction.

7 = = =

B. Exercises: Change mixed numerals to improper fractions and improper fractions to mixed numerals.

1. 1. 1

2. 2. 4

3. 2 3.

4. 5 4.

VI. Comparing Fractions

A. Examples: 7 > 4 means “7 is greater than 4” 3 < 5 means “3 is less than 5” 1. To compare fractions which have common denominators, compare numerators. The fraction with the greater numerator is the greater fraction. (You might think of greater as meaning bigger or larger.)

> since 4 is greater than 3.

2. To compare fractions which do not have common denominators, first rewrite the fractions with common denominators. Then compare numerators as we did in part 1. Compare and (The least common denominator (LCD) is 6.) = and = Since > , we conclude > . Thus is the greater of the two fractions.

3. Compare and (Note the LCD is 7  9 = 63.) = and = Since < , we conclude that < Thus is the greater of the two fractions.

B. Exercises: State which fraction is greater in each of these pairs. 1. or 1. Math 081 Review Materials 9

2. and 2.

3. and 3.

VII. Addition of Fractions A. Examples: 1. If fractions have a common denominator, add the numerators and retain the common denominator.

2. If the fractions do not have a common denominator, rewrite the fractions with a common denominator and proceed as in part 1.

+ = + = = 1

3. 2 2 + 1 = + 1 3 = 4 (note = 1 ) Note: Add the whole numbers and add the fractions. B. Exercises: 1. 1 1. + 2. 3. 7 2. + + 2 3. 10

VIII. Subtraction of Fractions A. Examples: 1. If the two rational numbers have a common denominator, subtract numerators and retain the common denominator.

– = =

2. If the two rational numbers do not have a common denominator, rewrite the fractions with a common denominator and proceed as in part a.

– = –

3. When subtracting mixed numerals, it is sometimes necessary to “borrow” (rename the rational numerals). 5 5 4 – 3 = – 3 = – 3 1 Since we could not subtract from , we renamed 5 as 4 + 1 = 4 .

B. Exercises: 1. – = 2. 1 1. – 2. 1

3. 12 4. 4 3. 3 Math 081 Review Materials 10

– 9 – 1 4. 2 IX. Multiplication of Fractions A. Examples: 1. When multiplying rational numbers, multiply numerators by numerators and denominators by denominators.

 = = 2. If possible, simplify fractions before multiplying by canceling common factors (divide by common factors). 2  =  = = 1 3. In the following example, cancel common factors two times. 3 1  =  =  = = 3 1 1

4. Rename mixed numerals as improper fractions before multiplying. 2 2  3 =  =  = = 7 1 B. Exercises: 1.  2.  1. 2. 3.  4. 4  1 3. 2 4. 5

X. Division of Fractions A. Examples: invert the divisor and multiply. invert divisor

1.  =  =

multiply

2. Use the rules for simplifying when multiplying:

1 1  =  =  =  = 1 2 1 1 3. If necessary, first convert mixed numerals to improper fractions.

1 1  7 =  =  =  = 1

B. Exercises: 1.  2.  1. 2. 1 3. 1  2 4. 3  4 3. 4. Math 081 Review Materials 11

XI. Decimals and Place Value A. Examples: 1. The place values of 534.3046 are illustrated below: 5 3 4 . 3 0 4 6

Hundreds Tens Ones Tenths Hundredths Thousandths Ten-thousandths 2. Convert a decimal to a fraction: 0.024 = Note the decimal had 3 digits after the decimal point and the equivalent fraction has 1 followed by 3 zeros in the denominator. 3. Convert a fraction to a decimal: = 0.0128 The fraction’s denominator has a 1 followed by 4 zeros in the denominator. The decimal has 4 digits after the decimal point.

B. Exercises: 1. In 620.173, the digit in the hundredths position is ______1. 7 2. Convert 0.123 to a fraction. 2. 3. Convert to a decimal. 3. 0.0049 XII. Comparing Decimals A. Exercises: 1. Which number is greater, 0.0007 or 0.00068? Line up the decimal points in the two numbers and then compare digit-by-digit (beginning at the decimal point).

Note 0.0007 = 0.00070 and 0.00068 = 0.00068 7 > 6, thus 0.0007 > 0.00068 2. True or False? 6.3006 > 6.390001 Note 6.3006 = 6.300600 and 6.390001 = 6.390001 0 < 9, thus 6.3006 < 6.390001 False

B. Exercises: 1. True or False? 2.001 > 2.0009 1. True 2. True or False? 0.0053 < 0.00521 2. False 3. Which number is greater, 23.001 or 23.0012 ? 3. 23.0012

XIII. Addition of Decimals A. Examples: 1. Add 2.76 and 5.162 Line up the decimal points and add as if they were whole numbers. Math 081 Review Materials 12

2.760 Note that 2.76 = 2.760 + 5.162 7.922

2. 86 + 0.51 + 0.6002 is illustrated below: 86.0000 Note 86 = 86.0000 0.5100 and 0.51 = 0.5100 + 0.6002 87.1102

B. Exercises: 1. 3.1 + 0.99 = 1. 4.09 2. 1.43 + 18 + 0.906 = 2. 20.336

XIV. Subtraction of Decimals A. Exercises: 1. Subtract 6.14 from 19.317 Line up the decimal points and subtract as if they were whole numbers: (“borrowing” if necessary).

19.317 – 6.140 13.177

2. 11 – 0.02 = 11.00 – 0.02 = 11.00 – 0.02 10.98

B. Exercises: 1. 4.123 – 0.23 = 1. 3.893

2. 3 – 0.012 = 2. 2.988 XV. Rounding Decimals A. Examples: 1. Round 2.176 to 2 decimal places. The answer will have 2 decimal places. Look at the digit in the 3rd decimal place. If that digit is 5 or greater, add .01 (Note .01 has 2 decimal places) Then discard all digits after the 2nd decimal place.

6 > 5, so add .01 to 2.176 then discard digits after the 2nd decimal place. The answer is 2.18

2. Round 7.817 to 1 decimal place.

1 < 5, so the answer is 7.8 Math 081 Review Materials 13

Do not add .1, just discard digits after 1 decimal place.

3. Round 14.2995 to the nearest thousandths

14.2995

5 > 5 so the answer is 14.300 Note that 14.2995 + .001 = 14.3005, then discard digits after 3 decimal places.

B. Exercises: 1. Round off 2.1572 to two decimal places. 1. 2.16 2. Round off 0.081 to one decimal place. 2. 0.1 3. Round off 7.119 to the nearest tenth. 3. 7.1

XVI. Multiplication of Decimals A. Examples: The number of decimal places in the answer will be the sum of the decimal places of the factors. Each digit to the right of the decimal point is counted as a decimal place.

1. Multiply 0.415 and 0.07 0.415 3 decimal places x 0.07 2 decimal places .02905 5 decimal places

2. Multiply 40.5 and 0.24

40.5 1 decimal place x 0.24 2 decimal places 1620 8100 9.720 3 decimal places B. Exercises: 1. 0.51 x 0.044 = 1. 0.02244 2. 7.012 x 2.11 = 2. 14.79532

XVII. Division of Decimals A. Examples:

1. 16 65.19 Round to two decimal places.

. Insert decimal point in quotient directly above 16 65.19 position in the dividend. Math 081 Review Materials 14

4.074 16 65.190 Perform division to one place beyond rounding 64 position. 1 19 1 12 70 Round answer: 64 4.074 becomes 4.07

2. .26 0.054 Round to two decimal places.

Decimal point moves two places to the .26 0.054 right in divisor. Thus decimal point moves two places to right in dividend.

. Insert decimal point in quotient above new 26. 005.4 place in dividend. Perform division to one place beyond rounding position. .207 26. 005.400 Round quotient: 52 .207 becomes .21 200 182

3. .8 28 Divide until there is no remainder.

.8 28. Place decimal point in dividend at end of whole number.

.8 28.0 Move decimal points in divisor and dividend. 35. Perform division. 8. 280. 24 40 Answer is 35. 40 Math 081 Review Materials 15

B. Exercises:

1. 8 16.16 1. 2.02

2. 28 116.95 Round to two decimal places. 2. 4.18

3. 68 275.4 3. 4.05

4. .76 1.536 Round to two decimal places. 4. 2.02

5. .25 75 5. 300

6. .4 11.38 Round to one decimal place. 6. 28.5

XVIII. Highest Common Factor (HCF) and Least Common Multiple (LCM). The highest common factor (HCF) of two numbers is the greatest number that will divide evenly into both numbers. The least common multiple (LCM) of two numbers is the least number which is a multiple of both given numbers.

A. Examples:

1. The HCF of 8 and 12 is 4 because 4 is the greatest number that divides evenly into both 8 and 12.

2. The HCF of 16 and 24 is 8.

3. The LCM of 8 and 12 is 24 because 24 is the least number that is a multiple of both 8 and 12.

4. The LCM of 16 and 24 is 48.

B. Exercises: 1. What is the HCF of 40 and 32? 1. 8 2. What is the HCF of 16 and 32? 2. 16 3. What is the LCM of 6 and 16? 3. 48 4. What is the LCM of 10 and 15? 4. 30

XIX. Addition of Numbers with Like Signs

A. Examples:

Rule: To add numbers having the same sign, take the sum of the numbers and give the result the common sign.

1. 3 + 8 = 11

2. -3 + (-8) = -11 Math 081 Review Materials 16

3. -17 -11 or -17 + (-11) + (-2) = -30 -2 -30

4. 15x + 7x = 22x

5. -6x + (-4x) = -10x

B. Exercises: Add

1. 15 + 7 = 1. 22

2. -10 + (-3) 2. -13

3. -12 + (-5) = 3. -17

4. -15 + (-20) + (-5) = 4. -40

5. -13 + (-46) = 5. -59

6. 7y + 3y = 6. 10y

7. 0 + (-5) = 7. -5

8. -6x + (-3x) = 8. -9x

XX. Addition of Numbers with Unlike Signs

A. Examples:

Rule: To add two numbers having unlike signs, take the arithmetic difference of the numbers and give the result the sign of the “greater” number.

1. 12 + (-7) = 5, since the arithmetic difference of 12 and 7 is 5, and the "greater" number, 12, is positive. 2. -17 + 6 = -11, since the arithmetic difference of 17 and 6 is 11 and the “greater” number, 17, is negative.

3. -10 + 3 = -7

4. -8 + 10 + 5 = 2 + 5 = 7

5. -11 + (-7) + 20 = -18 + 20 = 2

6. 3x + (-5x) = -2x

7. -10 + 10 = 0

8. 5 + (-5) = 0 Math 081 Review Materials 17

NOTE: In examples 7 and 8, we got 0 as answers because the numbers being added were opposites. Thus -2 + 2 = 0. Also 7 + (-7) = 0 because in each case, the numbers being added are opposites.

B. Exercises:

1. 18 + (-5) = 9. -15x + 5x = 1. 13 2. -2 + 7 = 10. Add -10x 2. 5 4x 3. 10 3. 15 + (-5) = 4. -9 11. Add 6y 5. 6 4. -12 + 3 = -6y 6. -4 7. -7 5. -75 + 81 = 12. 11x + -3x = 8. -15 9. -10x 6. -15 + 3 + 8 = 13. -9 + 9 = 10. -6x 11. 0 7. 4 + (-16) + 5 = 14. -9 and 9 are______. 12. 8x 13. 0 8. 6 + (-3) + (-18) = 14. opposites

XXI. Subtraction of Signed Numbers. A. Examples:

Rule: To subtract one number from another, change the sign of the subtrahend (the number being subtracted) and then add.

1. 10 – 7 = (change +7 to -7, then add) 10 + (-7) = 3

2. 15 – (-2) = (change -2 to +2, then add) 15 + 2 = 17

3. 15 – 17 = 15 + (-17) = -2

4. -8 – 5 = -8 + (-5) = -13

5. -15 – (-5) = -15 + 5 = -10

6. -2 – (-6) = -2 + 6 = 4

B. Exercises:

1. 50 – 13 = 6. 8x – 5x =

2. -13 – 50 = 7. -10 – (-5) =

3. 13 – 50 = 8. 45 – (-2) =

4. 3 – 12 = 9. -6a – (-4a) = 1. 37 2. -63 5. -12 – 5 = 10. 20 – (-3) =3. -37 4. -9 Math 081 Review Materials 18

11. 0 – 5 = 5. -17 14. -5 + (-3) – 2 = 11. -5 6. 3x 12. 4 12. 4 – 0 = 7. -5 15. 9 – (-6) = 13. -8 8. 47 14. -10 13. 6 – 10 + (-4) = 9. -2a 16. 1 – (-7) = 15. 15 10. 23 16. 8

XXII. Multiplication of Signed Numbers.

A. Examples:

Rule: The product of two numbers having like signs is always positive. [The product of two numbers having unlike signs is always negative.]

1. (2)(5) = +10 6. (-2)(-5)(-3)(-4) = (10)  (12) = 120 2. (-2)(5) = -10 7. (-3)(-y) = 3y 3. (-2)(-5) = +10 8. (-6)(-4)(x) = 24x 4. (2)(-5) = -10 9. (-2)(-2)(-2) = -8 5. (-2)(5)(-3) = (-10) (-3) = 30 10. (-5.1)(-3.5) = 17.85

B. Exercises:

1. (-3)(-5) = 2. -15 7. (-2)(4)(x) = 3. 15 2. (-3)(5) = 4. 30 8. (3)(0)(3) = 5. -126 3. (3)(5) = 6. 360 9. (-3)(-3) = 7. -8x 4. (-3)(-5)(2) = 8. 0 10. (3)(-9.9) = 9. 9 5. (-2)(-7)(3)(-3) = 10. -29.7

6. (-3)(-5)(-4)(-6) = 1. 15 XXIII. Division of Signed Numbers

A. Examples:

Rule: The quotient of two numbers having like signs is always positive; the quotient of two numbers having unlike signs is always negative.

1. = 9 or 27 ÷ 3 = 9

2. = +25 or (-50) ÷ (-2) = +25 3. = -5

4. = -20

5. = 0 Math 081 Review Materials 19

6. = undefined. In fact, any division by 0 is undefined.

B. Exercises: 1. = 2. -6 7. = 3. 9 2. = 4. -3 8. = 5. -8 3. = 6. -5 9. = 7. 3 4. = 8. -7 10. = 9. 0 5. = 10. undefined

6. = 1. 13 XXIV. Order of Operations

A. Examples:

Rule: Operations contained in parentheses or brackets are always performed first.

1. (2 + 5) + 3 = 2. (5.1 + 3.8) 2 = 3. [(2 + 7) + 5] – 3 = [Always start with the 7 + 3 = ( 8.9)  2 = [9 + 5] – 3 = innermost parentheses.] 10 17.8 14 – 3 = 11

If there are no parentheses to denote which operation should be performed first, perform multiplication and division first and then addition and subtraction. 4. 2 + 3  5 = 5. 2  3 – 4  5 + 3 = 2 + 15 = 6 – 20 + 3 = 17 -14 + 3 = -11 6. 8 ÷ 2 + 7 = 7. 8.2  7 + 1 = 4 + 7 = 57.4 + 1 = 11 58.4

Sometimes it is necessary to use the Distributive Law: a(b + c) = ab + ac

8. Simplify 6(x – 4) = 6(x + (-4)) = 6x + -24 or 6x – 24 NOTE: The 6x and -24 cannot be combined further.

9. Simplify -5(x + 2) = -5x + -10 or –5x – 10

B. Exercises:

1. 8 + (7 + 2  3) = 1. 21 2. 2  7 + 5 = 2. 19 3. (2 + 3)  (6 + 4) = 3. 50 4. 8 + 3  4 = 4. 20 5. 4(3x + 2) =5. 12x + 8 Math 081 Review Materials 20

6. 5(x – 3) = 6. 5x – 15 7. -2(2x + 5) = 7. -4x – 10 8. 2(x + 4) = 8. 2x + 8

XXV. Exponents A. Examples: 4 x means x  x  x  x \______/ 4 factors

3 1. 5 = 5  5  5 = 125 2. (-2)4 = (-2)(-2)(-2)(-2) = 16 4 4 3. -3 = -(3 ) = -(3  3  3  3) = -81

B. Exercises: 4 1. 5 = 1. 625 3 2. (-2) = 2. -8 2 3. (-4) = 3. 16 2 4. -4 = 4. -16 XXVI. Evaluating an Algebraic Expression

To evaluate an algebraic expression, replace the variable by its value and perform the operations.

A. Examples:

1. To evaluate 3 + x when x = 5, we get 3 + (5) = 8

2. Evaluate (2 + x) + 7 when x = 3: (2 + x) + 7 = (2 + 3) + 7 = 5 + 7 = 12

3. Evaluate 5 + x  y when x = 4 and y = 2: 5 + x  y = 5 + 4  2 = 5 + 8 = 13

4. Evaluate 4x2 when x = -1: 4(-1)2 = 4  1 = 4

5. Evaluate -x2 when x = 4: -(4)2 = -16

6. Evaluate 2xy - x when x = -2 and y = 3: 2(-2)(3) – (-2)= -12 + 2 = -10

B. Exercises: Evaluate the following:

1. x + 5 when x = 3 1. 8 2. 5(x + 4) when x = -1 2. 15 3. 3x when x = 5 3. 15 4. 3x – y when x = 2, y = -2 4. 8 5. a + (12 + b) when a = 3, b = -5 5. 10 6. a(5 + 2b) when a = 3, b = -1 6. 9 7. xy – 2y when x = -3 and y = 2 7. -10 3 8. 2x when x = -2 8. -16 2 9. 4x when x = -3 9. 36 Math 081 Review Materials 22

2 10. -x when x = -7 10. -49

XXVII. Simplifying Algebraic Expressions

A. Examples:

If parentheses immediately follow a minus sign, the parentheses may be removed using the distributive property. Each sign within the parentheses is changed.

1. 4a – (2a + 1) = 4a – 2a – 1 = 2a – 1

2. 6 – (-3x + 1) = 6 + 3x – 1 = 3x + 5 or 5 + 3x

3. 8x – (-2x – 3) = 8x + 2x + 3 = 10x + 3 or 3 + 10x

If parentheses immediately follow a plus sign, they may be removed without altering the signs of terms within the parentheses. If parentheses begin an algebraic expression and there is nothing in front of them, they may be removed without altering the signs of terms within the parentheses.

4. 4a + (2a + 1) = 4a + 2a + 1 = 6a + 1

5. (-3a + 2) + a = -3a + 2 + a = -2a + 2

Thus, to simplify an algebraic expression, remove all parentheses then combine like terms.

6. 2x + 5 – 3x + 2 = -x + 7

7. 3(x + 2) = 3x + 6 (by distributive property)

8. (3x – 1) – (2x – 3) = 3x – 1 – 2x + 3 = x + 2

9. 7(2x – 1) – 5 = 14x – 7 – 5 = 14x – 12 or -12 + 14x Math 081 Review Materials 23

10. -5(x – 2) – 3(x + 2) -5x + 10 – 3x – 6 = -8x + 4 or 4 – 8x

B. Exercises: Simplify each of the following.

1. (5 + x) – 8 = 1. x – 3 or –3 + x

2. -6(4x) = 2. -24x

3. -11x + x = 3. -10x

4. -(4 – 7x) = 4. -4 + 7x or 7x – 4

5. 4(3 – 2x) – (2 – 5x) = 5. 10 – 3x or -3x + 10

6. 2(x – 4) – 3(x + 4) = 6. -x – 20 or -20 – x

7. 1 + 6(-a + 1) = 7. -6a + 7 or 7 – 6a

8. a – 4(a – 1) = 8. -3a + 4 or 4 – 3a

9. -3(2x – 4) + 6 = 9. -6x + 18 or 18 – 6x

10. 7 – 2.5 (x – 4) = 10. -2.5x + 17 or 17 – 2.5x

XXVIII. Operations with Exponents

A. Examples: To multiply expressions with the same base, add the exponents. The rule is xa  xb = xa+b.

1. 3 2 5 x  x = x

4 5 2. x  x = x (Note: x = x1)

4 2 5 3 5 6 3. (3xy z)(4x yz ) = 12x y z

B. Exercises: 7 5 12 1. x  x 1. x

6 7 2. x  x 2. x

3 2 4 3 7 3. (2xy )(3x y ) 3. 6x y

2 5 7 4. (-2x )(4x ) 4. -8x

2 2 2 3 2 3 5 4 5. (-6x y z )(-4xy z ) 5. 24x y z Math 081 Review Materials 24

XXIX. Solving Linear Equations by adding the same quantity to both sides.

A. Examples:

1. Solve x + 7 = 12 (We add -7, the opposite of 7, to both sides to - 7 = -7 eliminate + 7 and thus isolate x.) x = 5 The solution is 5, because 5 is the number which when added to 7 equals 12.

2. Solve x – 5 = 8 5 = 5 (We add 5, the opposite of -5, to both x = 13 sides to eliminate -5 and thus isolate x.)

B. Exercises: Solve for x.

1. x + 2 = 9 1. 7

2. x – 13 = 5 2. 18

3. x + 5 = -6 3. -11

4. x – 3 = -3 4. 0

XXX. Solving linear equations of the type ax = b.

A. Examples: Solve for x. 1. Solve 8x = 72 x = 9 Note: "9" is the number that multiplied by 8 equals 72.

2. Solve -2x = -8 x = +4

3. Solve -x = 7 x = -7

1. 6x = 42 1. 7

2. -5x = 25 2. -5

3. -8x = -24 3. 3 Math 081 Review Materials 25

XXXI. Solving more complicated linear equations.

A. Examples: Solve using the set of integers.

1. -3x – 10 = 4x + 11 -4 x -4 x Add -4x to each side to eliminate the -4x -7x – 10 = 11 term on the right. + 10 = + 10 Add 10 to each side to eliminate the -7x = 21 constant term -10 on the left. x = -3

2. -x + 2 – 5x = 4x – 8 -6x + 2 = 4x – 8 Simplify each side first. -4 x -4 x -10x + 2 = – 8 - 2 = – 2 -10x = –10 x = 1

3. -3(x – 5) – 1 = 20 -3x + 15 – 1 = 20 Remove parentheses. -3x + 14 = 20 Simplify each side. -14 = -14 -3x = 6 x = -2

4. 3 – (x – 1) = 4(3 – x) – x 3 – x + 1 = 12 – 4x – x -x + 4 = -5x + 12 5x 5x 4x + 4 = 12 -4 = -4 4x = 8 x = 2

1. -2x + 11 = -4x + 3 1. -4

2. 7x – 7 = 8x – 8 2. 1

3. 3(2x – 1) – 2x = 9 3. 3

4. -6(2x – 3) – 5 = -11 4. 2

5. -x + 5 + 2x = 3(x – 1) 5. 4

6. -2 – (2x – 5) = -(x + 1) – 2 6. 6