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

11

2

4

1

114

21

22

1

44

2

A 5)

8

11

Textbook Page 80 When dividing fractions, multiplying by the reciprocal of the divisor.

11

2

4

1

2

11

4

1

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) )1(5

)8(8)2(12

15

6424

4

40

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 simplify before solving

24x –48 +48= 24x –48 +48 add 48

24x = 24x

24x – 24x =24x –24x sub 24x

0=0

Solutions are all real numbers.

True Statement

Special Equations II

More Example Solve the equation.

−2(3y − 5) = −6y + 1

−6y +6y+ 10 = −6y +6y+ 1 add 6y

10 = 1

There is no solution.

False statement

True statement False Statement

Solution: all real numbers 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

Ratio Sec 2.6

Practice test form A 16)

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 48

40 40

ft in

in in

6 8

5 8

6

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.

Practice form A 18 ) Complete table of values for the

equation 5x+y = –42. 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

5x+y = –42

5(1) + y = –42

5 + y = –42

–5 –5

y = –47

the ordered pair is (1, –47).

x y

–9

0

1 x y

–9 3

0 –42

1 –47

Completing a Table of Values

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) = (x1,y1) and (–7,2) = (x2,y2).

2 1

2 1

y ym

x x

2 ( 2)

7 ( 9)

2 2

7 9

4

2 2

Use () around negative numbers.

2 1 1 2

2 1 1 2

y y y ym

x x x x

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 = 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.

x y

0

0

x y

0 4

4 0

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.

Equation Graph

y= # Horizontal line

x = # Vertical line

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 2x3–6x2–x+10 when x = –2

2x3–6x2–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 = −22? (−2)2 = (−2)(−2) = 4 Answer: NO. −22 = −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

= 56

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

Practice form A 31) find the square.

(9m+2)2

=(9m+2)(9m+2)

=9m(9m)+9m(2)+2(9m)+2(2)

=81m2+18m+18m+4

=81m2+36m+4

Multiplying Binomials by the FOIL Method

Practice form A 32) decide whether the expression is positive, negative, or zero.

80+30

= 1 +1

=2

The expression is positive.

a 0 = 1 n

n

aa

1

am

an a

mn(a 0)

Practice form A 33) Perform the division. Write the answer with positive exponent.

Check: 4x2(2x4 –7x2)

=4x2(2x4) –4x2(7x2)

=8x6 –28x4

2

46

4

288

x

xx

2

4

2

6

4

28

4

8

x

x

x

x 2426 72 xx 24 72 xx

Never leave negative exponent in your final answer

Practice test A 34)

What polynomial, when divided by –6a2x5, yields 7a4x5+10x4+10a as a quotient.

Dividend=divisor × quotient

Answer= –6a2x5 (7a4x5+10x4+10a)

= –6a2x5 (7a4x5) –6a2x5 (10x4)–6a2x5 (10a)

= –42a6x10 –60a2x9 –60a3x5

axxaxa

101076

???? 454

52

Practice Form A 35) Perform the operation. Write the answer without exponents.

Separate numbers and 10

Quotient rule (sub exp)

Product rule (add exp)

no exponent

)1021()1013(

)107()1078(43

88

4

8

3

8

10

10

21

7

10

10

13

78

)4(838 103

1106

1211 10103

16

1211102

102 20