Review of Elementary Algebra Content 0 1 Table of Contents Fractions ...........................................................................................................................................1 Integers .............................................................................................................................................5 Order of Operations .........................................................................................................................9 Exponents .......................................................................................................................................11 Polynomials....................................................................................................................................18 Factoring ........................................................................................................................................22 Solving Linear Equations ...............................................................................................................31 Solving Linear Inequalities ............................................................................................................34 Graphing Linear Equations ............................................................................................................38 Finding the Equation of a Line ......................................................................................................43 Application of Linear Equations ....................................................................................................48 Functions ........................................................................................................................................50 Systems of Linear Equations .........................................................................................................53 Application of a System of Linear Equations ................................................................................62 Solving Linear Inequalities in Two Variables ...............................................................................64 Solving Systems of Linear Inequalities .........................................................................................67 Solving Absolute Value Equations ................................................................................................72 Solving Absolute Value Inequalities .............................................................................................76 Vocabulary Used in Application Problems....................................................................................79 Solving Application Problems .......................................................................................................81 Comprehensive Review of Elementary Algebra ............................................................................88 Revised May 2013 2i 3 FRACTIONS Lowest Terms Example: 1 Simplify: 12 4⋅ 3 = 15 5⋅ 3 1 4 = 5 Addition/Subtraction Rule: To add and subtract fractions you need a Common Denominator. Example: 2 3 23+ += 77 7 For this example, we already have the common denominator. 5 = 7 1 2 13 24 Example: + =⋅+⋅ 4 3 43 34 This example, we need to find a common denominator, or a denominator which is 38 = + DIVISIBLE by 3 and 4. One such denominator 12 12 is 12. 11 = 12 Example: 71713 −= −⋅ 12 4 12 4 3 73 = − 12 12 4 = 12 1 = 3 Note: Any number over itself is equal to 1. When you multiply a number by 1, you are not changing the number, just renaming it. 43 9 =1 =1 =1 43 9 1 FRACTIONS Multiplication/Division Rule: Multiplication and division of fractions do not require a common denominator. 3 Example: 7 12 7 12 ⋅=⋅ Cancel a common factor of 4 85 85 2 73⋅ = Multiply across 25⋅ 21 = 10 2 5 26 Example: ÷=⋅ 3 6 35 2 5 26 To divide by , multiply by the = ⋅ 6 35 6 1 reciprocal which is 22⋅ 5 = 15⋅ 4 = 5 2 FRACTIONS Problems Answers Perform the following operations. 75 1 1. + 12 12 21 13 2. + 35 15 31 5 3. − 43 12 55 5 4. + 6 12 4 51 23 5. − 76 42 71 1 6. − 12 2 12 3 11 7. + 2 4 4 4 11 8. 5 − 3 3 16 13 41 9. − 35 15 16 12 76 10. − 37 21 3 FRACTIONS Problems (continued) Answers Perform the following operations. 25 49 11. + 6 4 4 13 3 12. ⋅ 24 8 32 6 13. ⋅ 75 35 2 4 14. 2⋅ 15 15 32 9 15. ÷ 53 10 3 3 16. ÷ 2 8 16 53 35 17. ÷ 6 14 9 95 27 18. ÷ 43 20 4 INTEGERS Definition: The absolute value of a number is the distance between 0 and the number on the number line. The symbol for “The absolute value of a” is a . The absolute value of a number can never be negative. Examples: 1) −=44 2) 33= Addition of Signed Numbers: • Like signs: Add their absolute values and use the common sign. Examples: 1) −+−3( 2) =−( 32 + ) = −5 2) 48+=( 48 +) =12 • Unlike signs: Subtract their absolute values and use the sign of the number with the larger absolute value. Examples: 1) −+34 =+( 43 −) =1 2) 2+−( 6) =−( 62 − ) = − 4 5 INTEGERS Subtraction of Signed Numbers 1) Change the subtraction symbol to addition. 2) Change the sign of the number being subtracted. 3) Add the numbers using like signs or unlike signs rules for addition. Examples: 1) −−34 =−+− 3( 4) = −7 2) 6−−( 262) = +( ) = 8 3) −−−2( 4) =−+ 24 = 2 Multiplication and Division of Two Signed Numbers: • Like signs: The product or quotient of two numbers with like signs is positive. Examples: 1) (−3)( −= 4) 12 8 2) = 2 4 −16 3) = 4 −4 = 2 • Unlike signs: The product or quotient of two numbers with unlike signs is negative. Examples: 1) (−=−3)( 4) 12 −15 2) = −5 3 • Division by 0 is undefined. 3 Example: is undefined. 0 6 INTEGERS Addition and Subtraction of Signed Numbers Problems Answers Perform the following operations. 1. 53+−( ) 2 2. −+−62( ) –8 3. 7− 12 −5 4. −−11 4 −15 5. 6+ 2 +−( 13) −5 6. 85−−( ) 13 7. −−3( 4 − 11) 4 8. 51 4 −− − 62 3 9. (−9) +−( 14) + 12 −11 10. −−4.4 8.6 −13 11. 2+−−( 48) −10 12. 93 3 +− 10 5 10 13. −−8( −− 41) −( 92 −) 4 14. −+−83( ) +−+− 76( ) –24 15. −+4( − 12 + 1) −−−( 1 9) −5 7 INTEGERS Multiplication and Division of Signed Numbers Problems Answers Perform the following operations. 1. (34)(− ) –12 24 2. − 4 −6 3. (−−10)( 12) 120 0 4. 0 −2 3 10 5 5. −− 89 12 6. (−9.8) ÷−( 7) 1.4 −30 5 7. 28− 8. (−5.1)( .02) –.102 −40 − 4 9. 82−−( ) 10. (−−91)( − 2) −−( 6) 26 8 ORDER OF OPERATIONS Please Excuse My Dear Aunt Sally Parentheses ( ) Exponents Multiply Divide Add Subtract "as you see it, left to right" Brackets [ ] "as you see it, left to right" Square Roots Grouping Symbols Absolute Value Examples: Note: When one pair of grouping symbols is inside another pair, perform the operations within the innermost pair of grouping symbols first. 2 36÷( 5 − 2) +⋅ 6 7 54÷−⋅ 3( 2 3) −+ 12 3 2 36÷( 3) +⋅ 6 7 54÷− 3( 6) −+ 12 3 36÷+⋅ 9 6 7 54÷−[ 3] − 12 + 3 4+ 42 −−+18 12 3 46 −+30 3 − 27 Examples: Note: For a problem with a fraction bar, perform the operations in the numerator and denominator separately. 49− 7 +− 7 4⋅+ 2 7 8 + 7 15 = = = = 3 322−− 2 94 5 5 4 (6− 5) −−( 21) 14 −−( 21) 1 −−( 21) 1+ 21 22 = = = = = 2 −9 −− 3 422 − 9 −− 3 4 − 9 −− 3 16 27 − 16 11 ( )( ) ( )( ) ( )( ) 9 ORDER OF OPERATIONS Problems Answers Perform the following operations. 1. 5+− 35 2 18 2. 12÷+ 2 6 1 37 3. 43−+2 6 1 4. 62 −÷+ 8 2 4 12 5. −−+12( 4 3) −19 6. 5−− 2( 43 ) 5 −24 2 − 7. 92652 ÷−( ) 17 24− 1 8. 2 − (−+31) 5 +−2 − 19 9. 4 22( 5) 3 2 3 21 7 10. − 4 32 12 10 EXPONENTS Laws of Exponents Definition: an = aaa... a n times a = the base n = the exponent Properties: 1. Product rule: aam n= a mn+ Example: aa2 3= aa aaa = a23+ = a 5 2 times 3 times m 7 1 1 1 1 1 a mn− a aaaaaaa 75− 2 2. Quotient rule: = a Example: = =aa = n a5 aaaaa1 1 1 1 1 a m 2 3. Power rule: (aan) = nm Example: (a3) = aa 32 = 6 Raising a product to a power: Raising a quotient to a power: n n n nn aa (ab) = a b = bbn 00 4. Zero power rule: a0 =1 Example: ( x) =1;( 3 xx) = 300 = 1 1 = 1 Examples: (12ab2)( 3 a 3 b 5 c) = 36 a13++ b 25 c = 36 a 4 b 7 c 4 2x2 2 4 xx 24 16 8 = = 3 4 3 4 12 3y 3 yy 81 ++ + 3xy22( 2 xy+= 5 y2) ( 3 xy 22)( 2 xy) +( 3 xy22)( 5 y 2) = 6 x 2121 y + 15 xy 222 = 6 xy 33 + 15 xy 24 11 EXPONENTS Problems Answers Perform the following operations and simplify. 10 8 1. ( xy25)( xy 83) xy 99 2. (57ab6)(− ab 38) −35ab 3. xy10 4 xy43 x6 yz z 4. 10ab37 −5ab3 −2ab24 5. 423 3 12 6 (ab c ) ab c 6. 2 532 4 10 6 (−4xyz) 16xy z 7. 3 3 93 (−mn) −mn 8. 0 2 2 (3nx) 9x 9. 2303 4 3 12 (53x y) (− xy ) −27xy 3 6 10. − 3 −8x 4xy 3 2xyz z 12 EXPONENTS Negative Exponents Definitions: a−m 1 a−m = 1 am Note: The act of moving a factor from the 1 am = am numerator to the denominator (or from the a−m 1 denominator to the numerator) changes a negative exponent to a positive exponent. a−m bn −n m b a Examples: 10−4 1 1 10−4 = = 1 104 10,000 −− ab2 a 12 b a 1 a −2 ab−2 = = = Since b has a negative exponent, move 22 2 11bb it to the denominator to become b xy−−52 x 5 y 2 yz23 = Only the factors with negative exponents −3 = z z−3 x5 are moved. 13 EXPONENTS Example: −−22 Within the parenthesis, move the factors with −−12xy−23 12y3 = negative exponents. 42 242 33xy xxy −2 −4y = Simplify within the parenthesis. x6 −2 (−4) y−2 = −2 ( x6 ) Use the power rule for exponents. −2 (−4) y−2 = x−12 x12 = 2 Move all factors with negative exponents. (−4) y2 x12 = 16y2 Simplify. 14 EXPONENTS Problems Answers Perform the following operations and simply. 2 1. (−3) 9 2. −32 –9 3. 330⋅⋅− 12 3 8 11 = 34 81 −2 2 45⋅ −1 5 25 = 4. −3 26 2 4 2 1024 5. (33a04 b)( ab 32 c ) 9ab72 c 2 − 2 −13 18ac42 e 6. ( 32ebc) ( ab ) b11 − 32− 3 y6 7. ( xy ) x9 6ab02 c− −18ac2 8. −1 (−3a) bc−3 b 15 EXPONENTS Problems (continued) Answers 9. x−8 1 x8 10. −3 1 2 8 11. aaa−−4 10 8 1 a2 12. a−−6( aa 23) 1 7 a 13. x8 x12 −4 x 14. b−3 1 5 8 b b 15. a−4 a2 −6 a 16. p p6 −5 p 17. 4−5 1 −2 4 64 18. 28 8 5 2 19. 2−1 1 2 2 8 20. ab−4 1 53− 4 ab ab 21. xy−52 y5 −−23 3 xy x 16 EXPONENTS Problems (continued) Answers 22. bc−−28 1 42− 66 bc bc 23.