Summer Packet Final

ANSWER KEYS KEY Vocabulary/Know the difference Review 1. Proportions vs. multiplying fractions (cross multiplying for a proportion vs. multiplying numerators and multiplying denominators when multiplying fractions) Proportion Multiplying Fractions 2 8 2 8 = vs. ⋅ x 12 3 12 24 = 8x (2 ⋅ 8) 16 4 = = (3 ⋅ 12) 36 9 3=x 2. x * x = x2 vs. x + x = 2x 3. order of operations: don’t be tricked by these common “mistaken identities” a. 5 * 23 vs. (5 * 2)3 5 * 8 = 40 103 = 1000 b. 5 + 3(x + 4) 5 + 3x + 12 not 8(x + 4) c. -82 vs. -1 * 64 = 64 (-8)2 (-8)(-8) = 64 4. (x – 9)2 vs. (x + 9)2 vs. (x – 9)(x + 9) a. (x – 9)2 = x2 – 9x – 9x + 81 = x2 -18x + 81 b. (x + 9)2 = x2 + 9x + 9x + 81 = x2 +18x + 81 c. (x – 9)(x + 9) = x2 + 9x – 9x – 81 = x2 – 81 5. know the difference between a term, expression, equation, and inequality term – number, variable, or product of numbers and variables (ex: 2, x, or 2x) x expression – terms with mathematical symbols (ex: 2x, 2x – 3, x2 – 4x + 3, ) 2 equation – expressions set equal to one another (ex: x = 2, 4x + 3 = 12 – 5x) inequality – expressions not equal to one another (ex: x > 2, 4x + 3 ≤ 12 – 5x) 6. know the difference between solve, evaluate and simplify simplify – to rewrite an expression in simplest form possible where nothing else can be performed (includes no parentheses or negative exponents; all fractions have been reduced) evaluate – to find the value of (once the value has been found, the final result should be written in simplest form) solve – to work out the solution to the problem 7. know the difference between rational and irrational rational – real number that can be written as a fraction (ex:0, 1, terminating decimals] 1 , 121 , 0.25) [repeating or 3 irrational – real number that can’t be written as a fraction [nonrepeating, nonterminating decimals] (ex: 12 , π , 2.478192... ) 8. know the symbols by name: > as “greater than” vs. < as “less than” 9. know coefficient – numerical factor of a monomial [number being multiplied by a variable] 10. know factor – two or more numbers that multiply to produce another number [2 and 5 are factors of 10] 11. reduce entire factors not parts of a factor (individual terms) - you can reduce 5(2 x − 1) =5 2x − 1 - 10 x − 5 10 x − 5 by rewriting the numerator in factored form first: 2x − 1 2x − 1 = you can’t reduce 10 x − 5 10 x − 5 by trying to reduce just 2x − 1 2x − 1 Fractions 1) 2 4 10 + = 3 9 9 2) 7 4 19 − = 4 5 20 3) 3 1 11 + = 4 6 12 4) 2 4 8 • = 3 9 27 5) 7 4 7 • = 4 5 5 6) 3 1 1 • = 4 6 8 7) 2 4 3 ÷ = 3 9 2 8) 7 4 35 ÷ = 4 5 16 9) 3 1 9 ÷ = 4 6 2 DO NOT USE A CALCULATOR Solve Equations and Inequalities Review DO NOT USE A CALCULATOR x −3+ x 2= 1) + 7 = −5 x=-48 2) x=17 4 7 Clear out fractions. 7 1 3 1 3) 8 a − 4 + 4 a = 16 + a a=1/2 Variables on both sides. 4) 3( x – 8 ) + 3( 2x + 4 ) = 15 6) 6 – 2x + 5x = 7 + 7x – 15 x = − 7 2 x=3 5) 8 + 3( a – 3 ) = 4( a + 5 ) a = −21 Solve Inequalities. When you multiply or divide both sides by a negative remember to flip the inequality. Get the variable on the left side. − x + 6 > − ( 2 x + 4) 8) 2x − 6 < −x + 6 x > −10 x<4 4 -10 Word Problems You must be able to write the equation or inequality first. Then solve for the variable. 9) The greater of two numbers is 6 more than 4 times the smaller. Their sum is 41. Find the numbers. x + (4 x + 6) = 41 7 and 34 7) 10) Find three consecutive integers whose sum is 105. x + ( x + 1) + ( x + 2) = 105 34,35,36 11) Find three consecutive even integers whose sum is 138. x + ( x + 2) + ( x + 4) = 138 44, 46, 48 12) The length of a rectangle is 2 feet more than its width. If its perimeter is 40 feet, find the length and the width. 40 = 2w + 2(w + 2) Length: 11 Width: 9 13) The second angle in a triangle is 3° less than twice the first angle. The third angle measure 8° more than twice the first angle. Find each angle. x + (2 x − 3) + (2 x + 8) = 180 35, 67, 78 14) Jeffery has grades of 93 and 81 on the first two tests of the quarter. Progress reports will go home after the third test If Jeffery does not have an A average on his progress report, he cannot go to the football game that week. Jeffery will have to make at least what grade on the third test to be allowed to go to the football game? 93 + 81 + x ≥ 93 3 x ≥ 105 KEY Linear Equations Review: Slope, Writing Linear Equations, Horizontal & Vertical Lines, Parallel & Perpendicular Lines Find the slope (rate of change) of the following problems. 1. (3,−8) , (− 5,−1) − 2. Day Temperature ( F) 60 62 64 66 1 2 3 4 7 8 3. 3 2 2 4. The cost of museum tickets is $48 for four people and $78 for 10 people. What is the cost per person? $5 per person Write the equation of the line in slope-intercept form & standard form given the following. 5. ( 2,5 ) m=3 6. ( 4, 2 ) m=− 5 7 7. ( 2, −6 )(1, −2 ) 8. m = – 1 , b=2 2 y = 3x − 1 5 34 y =− x+ 7 7 y = −4 x + 2 1 y =− x+2 2 3x − y = 1 5 x + 7 y = 34 4x + y = 2 x + 2y = 4 9. (2, -3) m = 0 y = −3 10. (4, -1) m = undefined x=4 11. 3 y = − x +1 2 3x + 2 y = 2 12. The cost for 7 dance lessons is $82. The cost for 11 lessons is $122. Write a linear equation to find the total cost C for L dance lessons. Then use the equation to find the cost of 4 lessons. C = 10 L + 12 $52 13. Write the equation for a vertical line that goes through the point (2, 4). x=2 14. Write the equation for a horizontal line that goes through the point (-1, 3). y=3 Solve the equation for y if necessary, and find the slope. Then, find the slope of a line parallel and perpendicular to the original line. Slope Parallel Perpendicular 15. 2x + 6y = 8 1 4 y =− x+ 3 3 − 1 3 undefined/none= 16. x = 3 0 17. y = -2 − 1 3 undefined/none 0 3 0 undefined/none 18. Write the equation for the line parallel to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (3, –1) y= 4 x−5 3 4 x − 3 y = 15 19. Write the equation for the line perpendicular to the given line and through the given point. Write the answer in slope intercept and standard forms. 4x – 3y = 9 and (8, –3) 3 y = − x+3 4 3 x + 4 y = 12 20. Given a line through (–2, 4) and (8, –1), find the equation of the line perpendicular to that line through the midpoint of those points. Write the answer in slope intercept and standard forms. y = 2x − 9 2 4x − 2 y = 9 19) y = 35 x + 75 20) y = −7 x + 6791 21) y = 2 x + 10 Systems of Equations Review IF THE LINES INTERSECT ONCE, THE ANSWER IS THE ORDERED PAIR. IF THE LINES DO NOT INTERSECT (PARALLEL), THE ANSWER IS ∅. IF THE LINES ALWAYS TOUCH (ARE THE SAME LINE), THE ANSWER IS INFINITLY MANY SOLUTIONS. 1 3x − y = 5 y = − x+4 2) 1) 2 − x + 2y = 0 y = 2x − 6 (4, 2) 3) 3x + 4 y = 8 y=5 (2, 1) 4) (− 4, 5) 5) x+ y = 2 x + y = −1 ∅ 5x − 2 y = 4 x = −2 (− 2,−7 ) 6) 2x + 3 y = 9 4 x + 6 y = 18 infinitely many 7) x+y=4 y = 2x +1 8) (1, 3) 9) y = 2x − 5 3y − x = 5 (1, 2) 10) x = 3y − 4 2x − y = 7 12) − x − 2 y = −3 2x + 4 y = 6 infinitely many x = 2 − 2y x−y=6 x + y = −2 (2, − 4) (5, 3) 13) x = −2 y ∅ (4, 3) 11) x = y −1 y = 4 − 2x 14) 3x − 2 y = 0 4 x − 3 y = 15 (− 30, − 45) KEY Laws of Exponents Review (Remember: x a ⋅ x b = x a +b , ( x a ) b = x ab , (negative exponents should always be simplified) ( 1) 3a 4b −5a 7b3 ) 2) 6) − 40a −8 b 20 25a 6 b10 10) 8b10 − 14 5a 14) a7 b5 17) 2 2 4 ( −a b )( −b c )( −a c ) 3 2 3) ( −3x )( −2 x ) ( 5x ) 2 4 x 10 y 12 x10 y 7 (x ) 3 7 18) (a ) a 54 6 ( 3x ) 2 4 81x 8 9 22) ( 4x ) 64x 24 8 3 3 4 4) 4 0 1 7) (− ac )(− bc )(− ab ) 8) − 4 0 − a 2b 2 c 2 11) − 22 x 3 y 6 − 14 x13 y −3 −1 12) ( −4) 0 11y 9 7 x10 5 28a −8 b 2 21a −15 b10 2 2 − a 6b 4c 6 15) 4a 7 3b 8 x 21 21) 2 y a12b 2 a 5b 7 13) ( − y )( 3 y z )( −5 yz ) 30x 7 8c5 9) x 1 a −b 0 −a = x x = x = 1 , and ) xb xa 15 y 5 z 6 − 15a 11b 4 5) (2c 3 )(4c 2 ) a 1 −27 x3 y15 9 x13 y 6 16) 4 x 0 3y9 − 10 x 19) ( −2 x y ) 4 6 20) 64 x 24 y 6 23) ( −4 x y ) 5 7 4 3 − 64 x15 y 21 ( −2 x ) 4 3 − 8x 12 24) (−2 x 0 ) 5 − 32 Polynomials and FOIL 1) 30 x 3 + 15 x 2 − 10 x 2) 12a 2b6 − 20a 3b5 3) −12 x 4 − 24 x 3 + 6 x 2 4) x 2 + 11x + 24 5) 4 x 2 + 2 x − 12 6) 4 x 4 − 5 x 3 − 24 x + 30 7) 25 x 2 + 20 x + 4 Radical & Rational Exponent Review Simplify Radicals 1. 100 2. 36 6 10 6. 50 3. − 121 7. − 11 45 8. 3 5 2 7 11. 12. 3 98 13. 20 16. 20 x 2 y 17. 2x 5 y 21. 100a 2 18. 36x 2 (x + 5)12 −4 5 15 2 ( x + 5) 6 7x 2 15. 450 18a 2 3a 2 20 x 6 y 10 z 19. 75 x12 y 20 z 6 20. 2 x 3 y 5 5z x5 22. 2 2 x 7 72a 2 8 10. 14. 6a 2 10a 5. 9. − 80 6x 21 2 − 49 not real 28 5 2 400 4. 5 x 6 y10 z 3 3 23. 5 x 3 x2 x 5x x Rational Exponents See these as examples of rational exponents: Simplify the following. 24. Write as a rational exponent: 1 16 2 4 28. 121 25 1 2 1 125 3 = 3 25 = 25. 27 1 3 1 2 121 3 26. 256 29. 3 64 1 4 125 1 4 64 4 1 3 81 27. 30. 1 − 36 2 4 16 = 4 81 −6 16 1 4 KEY Function Notation Review State whether each set is a function. Answer yes or no. Find the domain and the range. 1) {(2, 5), (5, 6), (2, -6), (3, 8)} no 2) {(1, -2), (8, -4), (-3, 8), (-1, 2)} yes Domain: {2,3,5} Range: {− 6,5,6,8} Domain: {− 3,−1,1,8} Range: {− 4,−2,2,8} Use the vertical line test to determine whether each graph is the graph of a function. Answer yes or no. 3) no 4) yes 5) yes 6) no Use f ( x ) = x 2 − 3 and g ( x ) = 4 x − 1 to find each value. f ( −3 ) 7) g ( −7 ) 8) 6 11) f ( 3c ) 9c2 − 3 9) f g ( w − 7) 4w − 29 10) 11 − 9 − 29 12) 4 3 13) f ( 2m + 3) 4m2 + 12m + 6 f ( −5 ) + 8 30 14) −2 g ( x ) − 3 − 8x + 8 15) The temperature of the atmosphere decreases about 5oF for every 1000 feet increase in altitude. Thus, if the temperature at ground level is 77oF, the temperature t at a given altitude is found by using the equation t = 77 − .005h , where h is the height in feet. a) Write the equation in function notation where t is a function of h. [f(x) is meant as f is a function of x] t (h) = 77 − .005h b) Find t(100) and explain its meaning in this problem. 76.5 16) The function g ( x ) = 160 + 1.5 x models the weight gain of a basketball player as he starts a workout program where g is the weight in pounds after x weeks. a) Explain the meaning of 160 in the context of this problem. Starting/initial weight (y-intercept) b) Explain the meaning of 1.5 in the context of this problem. # of pounds added per week (slope) c) Evaluate g ( 6 ) and explain its meaning. 169; weight 6 weeks after starting Solve Quadratic Equations ANSWERS 1) ( x − 5)( x + 3) = 9 2) ( x − 8)( x + 1) = −20 x = 6,−4 x = 3,4 3) x( x + 1) = 72 4) ( x − 5) 2 + 4 x = 52 x = −9,8 x = 9,−3 5) (3 x + 2) 2 = 9 6) 2 x 2 + 56 = x 2 + 15 x 5 1 x=− , 3 3 7) ( x + 5)( x − 2) = 0 x = −5,2 x = 8,7