Unit 2 Real Number Line and Variables Examples Introductory Algebra Page 1 of Simplify. 22. Simplify

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1 Introductory Algebra Page 1 of 1 Questions 1 Find the additive inverse opposite) of 8 Find the absolute value 1 Add 6 + ) 4 Add 1 + ) 1 Add Add + 1 ) 4 7 Add 4 + 8) Aaron owes $8 to a credit card company He makes a purchase of $ with the card and then makes a payment of $10 What is his current balance on the credit card? 9 Explain in words who you would find 8 ) 10 Subtract 4 ) 11 Subtract Subtract 64 18) 1 Combine 64 7) + 14 Combine 4 0) 6 11) What is the temperature after a rise of 1 C from a start of 1 C? 16 Multiply and simplify ) 1 ) Multiply and simplify 18 Divide 6 9) 19 Divide 48 8) 0 Simplify 6 ) ) 4 ) 1 Simplify Simplify 8 1 ) ) 4 Evaluate ) 4 Evaluate Evaluate 16) 6 Evaluate 16 7 Evaluate ) 8 Evaluate 4) ) 9 Evaluate ) 4 0 Evaluate + 6) + 1 Evaluate 7) 6) ) 7 ) 8 ) ) Evaluate 6 ) + 8 Evaluate ) ) 4 4 Multiply using the Distributive Property 6 1x 4x + 18) Multiply using the Distributive Property x x + 10y 4) 6 Multiply using the Distributive Property x ) ) 7 Multiply using the Distributive Property 6x + y 1)x) 8 Multiply using the Distributive Property x + y 1) xy) 9 Multiply using the Distributive Property 1 x x + 4 ) y

2 Introductory Algebra Page of 1 40 The price of a cell phone was x A manager s special reduced the price by $ If the store sold 4y cell phones, use the distributive property to find the value of the cell phones sold 41 The Beverly Airport runway is 4x feet wide The airport was supposed to have a 000 ft long runway, however some of the land was too wet to build on so the length of the runway was decreased by y feet Use the distributive property to find the area of the runway 4 Combine like terms x + y 8x 7y 4 Combine like terms x x 7x + 8 x 44 Combine like terms 1 x y x y 4 Combine like terms 4 a 1 b 1 a 1 b 46 Combine like terms ab 7a 9ab + 4a 6b 47 Simplify ba b) + 4 ab b ) 48 Simplify 4 x) 1x) 49 Evaluate x x when x 0 Evaluate x 7x + when x 1 Evaluate a + ab b when a 1 and b A park has a triangular piece of land on the border The altitude of the triangle is 400 ft The base of the triangle is 80 ft What is the area of this piece of land? The radius of a circular opening on a chemistry flask is 4cm What is the area of the opening? 4 Find the total cost of making a triangular sail that has a base dimension of 1 ft and a height of 0 ft if the price of making the sail is $190 per square ft Dry ice is sold as carbon dioxide Dry ice does not melt, it goes directly from solid state to gaseous state Dry ice changes from a solid to a gas at 109 F What is this temperature in Celsius? 6 Rewrite x y using a negative sign and parentheses 7 Simplify 6x x y) 8 Simplify x[4x x )] 9 Simplify 6aa a 4) aa ) 60 Simplify a 4[b bb + )] 61 Simplify {x + 4[x x)]} 6 Simplify 4{a [4a b + a )]} 6 Simplify {x [x x x )]} 64 The base of an office building is in the shape of a trapezoid The altitude is 400 ft, and the bases are 700 ft and 800 ft What is the area of the base of the office building? If the base has a marble floor that costs $ per ft, what was the cost of the floor? 6 Replace the? with either < or > in? 6 66 Replace the? with either < or > in 8? 67 Graph the inequality on a number line x 68 Graph the inequality on a number line x > 69 Convert the inequality 4 < x < 10 to interval notation 70 Convert the inequality / x to interval notation 71 Convert the inequality x > 0 to interval notation 7 Convert the inequality 1x < 6 to interval notation 7 Convert the inequality 7 > x to interval notation

3 Introductory Algebra Page of 1 Solutions ) ) 1 + ) ) ) $8 + $ $10 $ First, change subtracting negative three to adding positive three Then use the rules for adding two real numbers with different signs 8 ) ) subtract by adding the opposite) common denominator) simplify)

4 Introductory Algebra Page 4 of ) ) ) 6 11) C + 1 C 8 C 16 ) 1 ) 11 ) ) multiplying two negatives gives a positive) ) ) 1 17 Multiply and simplify ) ) ) ) multiplying a negative and a positive gives a negative) ) ) dividing two negatives gives a positive) dividing two negatives gives a positive)

5 Introductory Algebra Page of ) 6) 7 6) 7 ) ) instead of dividing, invert divisor and multiply) multiplying two negatives gives a positive) 1 Simplify ) ) ) 1 ) 8 ) ) instead of dividing, invert divisor and multiply) multiplying a negative and a positive gives a negative) ) 7 ) 8 ) ) ) ) ) ) overall sign is positive) ))) 7 ) ) ) ) ) 16) 16) 16)16) )16) 6 7 ) ))))) ) ) )) 9 8 4) ) 4) 4) 4) ) ) 4)4)4))) 76

6 Introductory Algebra Page 6 of 1 9 Order of Operations ) 4 ) 4 brackets) 0 Order of Operations 9 4 powers) 4 multiplication/division from left to right) 1 multiplication/division from left to right) + 6) + 4) + brackets) 1 Order of Operations 8 + multiplication) 1 addition) 7) 6) ) 6) brackets) Order of Operations 4) 6) powers) 1 18 multiplication) 6 subtraction) 6 ) ) + 8 brackets) powers) multiplication) 4 addition/subtraction) ) )

7 Introductory Algebra Page 7 of x 4x + 18) 6 1x 6 4x distributive property) 1x 4x simplify, multiplication of fractions) 6x 6 4x simplify, prime factor and cancel) 10x 0x + 1 simplify, multiplication) x x + 10y 4) x x + x 10y x 4 distributive property) x x + x 10y x 4 simplify, multiplication of fractions) x + xy 4x simplify, prime factor and cancel) x + xy 4x 6 x ) ) x ) ) distributive property) 4x + 6 simplify, multiplication) 7 6x + y 1)x) 6x x) + y x) 1 x) distributive property) 18x + xy x simplify, multiplication) 8 x + y 1) xy) x xy) + y xy) 1 xy) distributive property) xy xy + xy simplify, multiplication) 9 1 x x + 4 ) y 1 x x + 1 x 4 y 1 x distributive property) 1 x x x xy 10 x + 1 x 4 y 1 x simplify, multiplication) 40 value sold number sold)original price - price reduction) 4y)x ) 4y)x) 4y)) 8xy 0y dollars

8 Introductory Algebra Page 8 of 1 41 area length)width) 4x)000 y) 4x)000) 4x)y) 1000x 8xy ft 4 Underline the like terms using a different style for each Remember to include the sign! 4 x +y 8x 7y x y If you need to include extra steps to see the simplification do so: x +y 8x 7y x 8x+y 7y rearrange so like terms are together) 8)x + + 7)y use distributive property in reverse which is factoring)) x y simplify) x x 7x+8 x x x x 7x + 8 1)x + 7)x + + 8) 1)x + 10)x + +) x 10x x y x+4 7 y 1 x x y y 1 ) x ) y 7 1 ) 7 x ) y ) 14 x ) y 1 ) ) x + y 1 1 ) ) 1 x + y x 1 y

9 Introductory Algebra Page 9 of a 1 b 1 a 1 b 4 a 1 a 1 b 1 b 4 1 ) a ) b ) 1 a ) a ) b 6 ) ) 1 4 a + b 0 6 ) ) 11 a + b a 6 b ) b 46 The underlining is not required if you can do the arithmetic in your head but make sure you get the arithmetic done correctly!) ab 7a 9ab + 4a 6b 8ab a 6b ba b) + 4 ab b ) b)a) b)b) + 4 ab) 4b ) distribute) 1ab + 9b 1ab 0b multiplication) 1 1)ab + 9 0)b collect like terms) 7ab 11b simplify) 4 x) 1x) 4 4x + ) ) )1x) distribute) 8 4x x 8 + 1) + 4x + 6x) )x + x 49 Write the equation using brackets where there are x: value ) ) ) ) put in the brackets) 9) + 1 simplify)

10 Introductory Algebra Page 10 of 1 0 Evaluate x 7x + when x value ) 7 ) + ) 7) If you have trouble doing two variables, do it in two steps value ) + )b b 1) + 1)b b 1 b b 1 ) ) 1 ) ) ) ) do a first) put in a 1) simplify) now do b) put in b ) simplify) cancel common factor) 1 Include the units Area of triangle 1 base)perpendicular height) 1 80 ft)400 ft) 6000 ft Area of Circle πradius) π4 cm) π16 cm ) 16π cm 0 cm

11 Introductory Algebra Page 11 of 1 4 First, find the area of the sail Area of triangle 1 base)perpendicular height) 1 1 ft)0 ft) 10 ft Total Cost area)cost per square ft) 10 ft )$190 1 ft ) $40 T C 9 T F ) 109 ) 9 78 C 6 x y x + y) x x y) 6x + )x) )y) distribute) 6x x + 6y multiply) x + 6y collect like terms) x[4x x )] x[4x x ))] distribute) x[4x x + 6] simplify,multiplication) x)4x x)x + x)6 distribute) 8x 4x + 1x multiply) 6aa a 4) aa ) 1a 18a 4a a + a 1a 18a 4a a +a 1a 19a a a 4[b bb + )] a 8b + 1bb + ) a 8b + 1b + 4b a + 1b + 16b

12 Introductory Algebra Page 1 of {x + 4[x x)]} 6x + 8[x x)] distribute the ) 6x + 8[x x)] no change, just highlighting number) 6x + 16x 8 x) distribute the 8) 6x + 16x 8 x) no change, just highlighting number) 6x + 16x 4 + 8x distribute the 8) 6x + 4x 4 collect like terms) 4{a [4a b + a )]} 1a + 8[4a b + a )] distribute the 4) 1a + 8[4a b + a )] 1a + a 8b + a ) distribute the 8) 1a + a 8b + a ) 1a + a 8b 8a distribute the 8) 1a 8b collect like terms) {x [x x x )]} x + 6[x x x )] distribute the ) x + 6[x x x )] x + 6x 6x x ) distribute the 6) x + 6x 6x x ) x + 6x 6x + 1x distribute the 6) 10x collect like terms) 64 Area of trapezoid 1 altitude)[base 1) + base )] ft)700 ft ft) 00 ft)100 ft) 00, 000 ft Cost of marble floor area)cost per square ft) 00, 000 ft )$ 1 ft ) $16, 00, > 6 since 6 is to the left of on the number line

13 Introductory Algebra Page 1 of 1 66 To compare fractions, we need a common denominator The common denominator is So we see 8 < 67 Note the closed circle, since we are including the endpoint in the inequality Note the open circle, since we are not including the endpoint in the inequality , 10) 70, /], with square bracket since / is included 71 0, ) 7, 1) 7, 7)