MATH97 Testing Enhancement Workshop Department of Mathematics and Computer Science Coppin State University Dr. Min A Updated on Oct 20, 2014
Addition and Subtraction Same sign: Ex: 1+3 = 4 1 3 = 4 Keep sign and add the absolute values. Different signs: Ex: 1 + 20 = +19 = 19 11 29 = 18 Give the sign of the number having the larger absolute value and subtract absolute values.
Multiplication and Division Same sign: positive Ex: -1 (-3) = +3 = 3-18 (-3) = +6 Different signs: negative Ex: -1 (20) = - 20 1 2 4 11 1 2 411 2 44 1 22
A 5) 1 2 4 11 1 11 4 2 Textbook Page 80 When dividing fractions, multiplying by the reciprocal of the divisor. 11 8
Select the lesser of two numbers 2 = 2 = 2 2 = 1 2 = 1 2= 2 2 = 1 2 = 1 2 = 2 A2) 2 and 20 20 = 20 20 is the lesser; B6) 19 and 23 19 =19; 23 =23; 19 is the lesser.
Evaluate sighed number expressions A7) evaluate ( 6x 3y)( 2a) given x= 2, y=3 and a= 4. Substitute x, y and a by the given numbers. ( 6x 3y)( 2a) = [ 6( 2) 3(3)][ 2( 4)] = (12 9)(8) =3(8) =24 Always use parenthesis around the negative numbers.
Perform the indicated operations A8) 12(2) 8(8) 5 ( 1) 24 64 5 1 40 4 10
The difference A4) After one round in a card game, your score was 44 points. After the second round, your score was 42 points. How many points did you lose in the second game? Gain +/ Lose 42 44 = (44+42)= 86
The difference of signed numbers B9)The stock market gained 15 points on Tuesday and lost 11 points on Wednesday. Find the difference between these changes. 15 ( 11) = 15+11 =26
Deciding Whether an Ordered Pair is a Solution Practice Form A 1) which pair of values of x and y makes 4x+y equal 10? Search the solution of the equation 4x+y = 10. x= 1 and y = 6 4(1) + 6 = 10 4+ 6 = 10 true The solution in ordered pair is (1,6). Practice form B 27) is (7, 8) a solution of x+y = 10? Substitute x by 7 and y by 8 and rewrite equation. 7+8 = 10 15 = 10 No, it is not a solution.
Special equations I Practice Form A 13) 24(x 2) = 6 (4x+3) 66 24x 48 = 24x+18 66 clear () 24x 48 = 24x 48 24x 48 +48= 24x 48 +48 add 48 24x = 24x 24x 24x =24x 24x 0=0 True Statement simplify before solving sub 24x Solutions are all real numbers.
Special Equations II More Example Solve the equation. 2(3y 5) = 6y + 1 6y +6y+ 10 = 6y +6y+ 1 add 6y 10 = 1 False statement There is no solution. True statement Solution: all real numbers False Statement There is no solution.
Consecutive numbers Sec 2.4 Practice form A 14) Two pages that face each other in a book have 485 as the sum of their page numbers. What is the number of the page that comes first? Idea: two numbers are consecutive. Let x = the smaller; x+1 = the larger. x+ (x+1) = 485 2x = 484 x = 242
Practice test form A 16) Ratio Sec 2.6 Express the phrase as a ratio in lowest terms: 4 feet to 40 inches. first convert 4 feet to inch. 4 ft = 4 (12) = 48 in The ratio of 4 feet to 40 inches is thus 4 ft 48in 40in 40in 68 6 58 5 Caution: common units first
Quadrants Section 3.1 Practice Form A 17) T or F? In quadrant IV, the y-coordinate is always positive. Practice Form B 30) T or F? The x-coordinate is positive in quadrant I and IV.
Completing a Table of Values Practice form A 18 ) Complete table of values for the equation 5x+y = 42. x 9 0 y 1) Let x = 9 5x+y = 42 5( 9) + y = 42 45 +y = 42 +45 +45 y = 3 the ordered pair is ( 9, 3) 2) Let x = 0 5x+y = 42 5(0) + y = 42 y = 42 the ordered pair is (0, 42). 3) Let x = 1 1 x y 5x+y = 42 5(1) + y = 42 5 + y = 42 5 5 y = 47 the ordered pair is (1, 47). 9 3 0 42 1 47
Completing Ordered Pairs Practice form A 19) Complete each ordered pair for y = x+2 a) (0, ) Substitute 0 for x & solve for y. y = x+2 equation y = 0+2 x = 0 y = 2 solve for y the ordered pair is (0, 2). b) (, 0) Substitute 0 for y & solve for x. y = x+2 equation 0 = x+2 y = 0 + x + x collect x x = 2 The ordered pair is (2, 0 ). c) (1, ) Substitute 1 for x & solve for y. y = x+2 equation y = 1+2 y = 1 The ordered pair is (1,1).
Practice form A 20) Find the sl0pe of the line through the pair of points ( 9, 2) and ( 7,2). Let ( 9, 2) = (x 1,y 1 ) and ( 7,2) = (x 2,y 2 ). y y m x x 2 1 2 1 2 ( 2) 2 2 4 2 7 ( 9) 7 9 2 Use () around negative numbers. y y y y m x x x x 2 1 1 2 2 1 1 2 Be consistent with order.
Deciding Whether a Given Ordered Pair is a Solution Try practice form A 23) Is (4, 5) a solution of the system x + y = 1 x y = 9? Substitute 4 for x and 5 for y in each equation. Eq1: 4 + 5 = 1? 9= 1? False Eq2: 4 5 = 9? 1= 9? False (4,5) is not a solution of this system because it does not satisfy equations.
Practice form A 24) solve the system 7x+6 = 4y 5x+2y = 6 Rewrite in Ax+By = C 7x+4y = 6 eq1 5x+2y = 6 eq2 Multiply eq2 by 2 2(5x+2y)= 2( 6) 10x 4y = 12 Group two equations with opposites 7x+4y= 6 10x 4y = 12 Add equations 3x = 6 x = 2 div 3 both sides Pick any equation to solve y. 5x+2y = 6 5 ( 2)+2y = 6 10 + 2y = 6 + 10 + 10 2y = 4 y = 2 Write the solution with ordered pair. The solution is ( 2,2). Check
practice form A 25) Solve the system of equations by graphing both equations on the same axes. x+y = 1 x y = 11 x+y = 1 y = x 1 : slope 1, y-int (0, 1) rise/run = 1/1 x y = 11 y= x +11: slope 1, y-int (0,11) rise/run = 1/1 The solution is ( 6,5).
Solve the system by the elimination method. x+y = 1 + x y = 11 add two equations 2x = 12 +y y=0 x = 6 divide both sides by 2 Sub 6 for x in eq1 x + y = 1 6 +y = 1 y = 5 add 6 both sides The solution is ( 6,5).
Practice Form B 29) Find the intercepts for the graph of x + y = 4. Then draw the graph. Idea: let x = 0 in the given equation and solve for y. let y = 0 in the given equation and solve for x. x y 0 0 x y 0 4 4 0 x = 0 y = 0 x+y = 4 x+y = 4 0 + y = 4 x +0 = 4 y = 4 x = 4 y-int (0,4) x-int (4, 0) Graph the line.
Equation Graph y= # Horizontal line x = # Vertical line Practice form A 22) if the y term is missing in both of two linear equations, the lines are examples: 1) x = 3, x= 2 two vertical lines, they are parallel. 2) x= 3, x = 3 same line, they are overlapped.
Recall The monetary value (in dollars) of x dimes $0.10 x The monetary value (in dollars) of y quarters $ 0.25 y Practice Form A 26) The monetary value (in dollars) of x dimes and y quarters $0.10x + $0.25y
Evaluate the polynomials Practice Form A 27) Evaluate 2x 3 6x 2 x+10 when x = 2 2x 3 6x 2 x+10 = 2( 2) 3 6( 2) 2 ( 2)+10 =2( 8) 6(4)+2 +10 = 16 24+12 = 40+12 = 28 Use parentheses to avoid errors Question: ( 2) 2 = 2 2? ( 2)2 = ( 2)( 2) = 4 Answer: NO. 2 2 = 1(2 2) = 1(4) = 4 ( 2) 3 = ( 2)( 2)( 2) = 8
Practice form A 29) write the expression using exponents. 5 5 5 5 5 5 6 = 5 6 Practice form A 28) Simplify the expression. ( 8)( 8) 2 ( 8)( 8)( 8) 4 =( 8) 1 ( 8) 2 ( 8) 1 ( 8) 1 ( 8) 4 =( 8) 1+2+1+1+4 =( 8) 9 Another way: ( 8)( 8) 2 ( 8)( 8)( 8) 4 =( 8) ( 8)( 8) ( 8)( 8) ( 8)( 8)( 8)( 8) = ( 8) 9
Distributive Property 2(5) = 10 2(5+x) = 2(5) + ( 2)x= 10 2x A10) Use the distributive property to rewrite the expression. 8(3x) 8( 5y) = 8(3x) + ( 8)( 5y) = 8(3x 5y)
Practice Form A 30) 6( 11x+2) =6( 11x)+6(2) = 66x +12
Multiplying Binomials by the FOIL Method Practice form A 31) find the square. (9m+2) 2 =(9m+2)(9m+2) =9m(9m)+9m(2)+2(9m)+2(2) =81m 2 +18m+18m+4 =81m 2 +36m+4
a 0 = 1 a n a Practice form A 32) decide whether the expression is positive, negative, or zero. 1 n a m a n amn (a 0) 8 0 +3 0 = 1 +1 =2 The expression is positive.
Practice form A 33) Perform the division. Write the answer with positive exponent. 8 6 x 8 4x 28x 2 4x 6 x 2 4 28x 4x 2 4 6 2 4 2 2 4 2 x 7x 2x 7x Check: 4x 2 (2x 4 7x 2 ) =4x 2 (2x 4 ) 4x 2 (7x 2 ) =8x 6 28x 4 Never leave negative exponent in your final answer
Practice test A 34) What polynomial, when divided by 6a 2 x 5, yields 7a 4 x 5 +10x 4 +10a as a quotient.???? 4 5 7a x 10x 4 5 6a x 2 Dividend=divisor quotient 10a Answer= 6a 2 x 5 (7a 4 x 5 +10x 4 +10a) = 6a 2 x 5 (7a 4 x 5 ) 6a 2 x 5 (10x 4 ) 6a 2 x 5 (10a) = 42a 6 x 10 60a 2 x 9 60a 3 x 5
Practice Form A 35) Perform the operation. Write the answer without exponents. 8 (7810 ) (710 3 (1310 ) (2110 8 4 ) ) 78 13 10 10 610 1 3 8 3 83 7 21 1 3 11 6 10 210 1112 210 20 10 10 10 10 12 8 4 8( 4) Separate numbers and 10 Quotient rule (sub exp) Product rule (add exp) no exponent