MAT 1033C Final Exam Review BY: Math Connections/Hands-On Math Table of Contents Useful Formulas Rational Expressions/Equations #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49
Radicals and Rational Exponents/Equations #10, 11, 12, 13, 14, 15, 16, 17, 18, 19 Complex Numbers #20, 21, 22, 23 Quadratic Equations/Applications #24, 25, 26, 27, 28, 29, 30, 31, 32 Graphing
#33, 34, 35, 36, 37, 38, 39, 40, 41 Equations of Lines/Slope #42, 43 Inequalities (Solving/Graphing) #44, 50, 51, 52, 53, 54, 55 Systems of Equations/Applications #45, 46
Other Topics to Study Study and Test Taking Tips Helpful Formulas to Remember Slope/Linear Equations m= Slope-intercept form: y = mx + b Function Notation: f(x) = mx + b Point-slope form: y y1 = m(x x1) Parallel Lines: Same Slope Perpendicular Lines: opposite, reciprocal slopes
Work Formulas Set-Up: (3 people) x= Formula: Factor: a3 + b3 OR a3 b3 Formula: (a b)(a2 ab M O
Match Opposite b 2) P Plus a and b are cube roots! a =2b x 8
3 (a b)(aa ab bb) (x 2) (xx + 2x + 22) M O P Match Opposite Plus
(x 2)(x + 2x + 4) 2 Problem #1 (Rational Expressions) Factor & Cancel like factors! 9x4 72x 9x(x3 8) 9x(x 2)(x2 + 2x + 4) 3x 12 3(x 4) 3(x 2)(x + 2) x2 + 1x 2 (x + 2)(x 1) 4x3 + 8x2 + 16x 4x(x2 + 2x + 4) 2 2
Problem #1 CONT 9 ( )( + +) ( +)( 1) 3 ( ) ( +) 4 ( + + ) 3 9( ) 34
3( ) 4 Problem #2 (Rational Expressions) Factor & Cancel like factors! x2 + 13x + 36 (x + 9)(x + 4) x + 14x + 45 (x + 9)(x + 5) 2 x2 + 5x x(x + 5) x2 3x 28 (x 7)(x + 4)
Problem #2 CONT ( +)( + ) ( +) ( +)( +) ( )( +) ( ) Problem #3 (Rational Expressions)
Division Multiply by the reciprocal. Keep, Change, Flip! Always flip the 2nd fraction! + + + + + Problem #3 CONT
Factor & Cancel like factors! + + + + + x2 + 5x 6 (x + 6)(x 1) x + 9x + 18 (x + 6)(x + 3) 2
x + 7x + 12 (x + 4)(x + 3) x2 1 (x 1)(x + 1) 2 Problem #3 CONT + + + + + ( +)( ) ( +) ( +)
( +)( +) ( +)( ) ( +) ( +) + +1 Problem #4 (Complex Fractions) Find LCD and multiply each term by LCD!
3 108 +36 2 3 + LCD = 12x 3 9 + 12x 12x 1 12x+ 12x
4 12 = Problem #5 (Complex Fractions) Find LCD and multiply each term by LCD! LCD = x2 4
x5 2 + 2 x x 2 x2 252 16 x2 x +
Problem #6 (Complex Fractions) Find LCD and multiply each term by LCD! LCD = x2 9 x1 2 + 2 x x 2
7292 2 x+ 2x + 9 + Sum of Cubes Factor!
Formula: Factor: a3 + b3 OR a3 b3 Formula: (a b)(a2 ab M O Match Opposite b 2)
P Plus a and b are cube roots! a =9b x + 729 3 (a b)(aa ab bb) (x + 9) (xx 9x + 99) M
O P Match Opposite Plus (x + 9)(x 9x + 81) 2 Problem #6 CONT + 9
+ = + Problem #7 (Complex Fractions) Find LCD and multiply each term by LCD! NOTE: 11 x = (x 11) 10 11
+ 11 3 8 + 11 Problem #7 CONT LCD = x(x 11) 10 11 x(x 11) + x(x 11) ( )
( ) 3 8 x(x 11)+ x(x 11) ( ) = Problem #7 CONT =
= Problem #8 (Polynomial Division) (x 8) 4x2 33x + 8 Long Division of polynomials! Steps on next slide! 4x -1 4x = 4x2 x ____ 2
4x(x 8) = 4x 32x (x 8) 4x 33x + 8 Change signs! 4x2 + 32x -1 x ____ = -1x 1x + 8 + 1x 8 -1(x 8) = -1x + 8 2
Remainder = 0 4x - 1 Change signs! Problem #9 (Polynomial Division) 3 2 15x + 31x 2x 17 (3x + 5)
Long Division of polynomials! Steps on next slide! 5x2 +2x 4 (3x + 5) 15x3 + 31x2 2x 17 15x3 25x2 6x2 2x 17 6x2 10x 12x 17 + 12x + 20 Remainder = +3
5x2 + 2x 4 + 2 5x 3x ____ = 15x3 5x2(3x + 5) = 15x3 + 25x2 Change signs! +2x = 6x2 3x ____ 2x(3x + 5) = 6x2 + 10x Change signs!
-4 = -12x 3x ____ -4(3x + 5) = -12x 20 Change signs! Problem #10 (Rational Exponents) Find LCD: LCD for 3, 5, and 2 = 30 4 3
6 5 1 2 Problem #10 CONT 4 3
6 5 1 2 40
Properties to Remember: 1) xm xn = xm + n 2) + ( )=
36 15 Problem #11 (Rational Exponents)
= = Add exponents! xm xn = xm + n Subtract exponents! 20 1 = = 6 6
Problem #12 (Radicals) 6k q 3 4
6k q 3 4 Problem #13 (Radicals) = = 3x y 2
2 3x y Problem #14 (Radicals) = = = = NOTE:
=3 =2 =6 =9 Problem #15 (Radicals) = = = = NOTE: =3 Since 333 = 27
=4 Since 444 = 64 Problem #16 (Radicals) Isolate radical! 2 1+=10 4 2 2 1= 2x 1 +1 +1
4 2x 2 2 2 x Problem #16 - - Checking: 2 1+ 4=10
? 2 1+4 =10 ? +4=10 +4=10 + 4=10 x Problem #17 (Radicals) 2
31 =( 1) 2 Square both sides! =( )( ) FOIL! equation = + Make equal to zero! 31 + x
+ 1x 31 0 Problem #17 CONT 0 0+ 5 6 30 Solve: x 6 = 0; x + 5 = 0 +6
+6 5 Factor trinomial! 5 1 30 2 15 3 10 Solutions: x = 6; x = 5 5 6
Problem #17 - - Checking! x = 6; x = 5 1 31 = ? 31 = 1 25=5 ? 31( )= 1 36= Extraneous Solution
x=6 Problem #18 (Radicals) Square both 2 sides! 2 2 +5=(3+ 2) 2 +5= FOIL! 9
+ + +x2 Problem #18 CONT +x2 +x x 7 Isolate x 7 2
FOIL! x2 4x + 4 = 36x 72 2 Distribute! Problem #18 CONT. x2 4x + 4 = 36x 72 36x +72 36x x 40x + 76 = 0 (x 38)(x 2) = 0 2
x = {2, 38} Check your solutions! +72 Make equation equal to zero! Solve by factoring! +76 1 76 2 38 4 19
Problem #18 - - Checking! x = 2; x = 38 2 +5=3 + 2 2 +5=3+ 2 9=3+ 0 3=3 2 +5=3+ 2 81=3+ 36 9=3+ 6 NO Extraneous Solutions! Problem #19 (Radicals)
2 2 4 +5=( 2 2 3) Square both sides! +=( 3)( 3) FOIL! += +
+= + Problem #19 CONT += + += +7 Isolate -6 2
( )=[ ] 2 Square Both Sides! ( )( )= ( ) FOIL! Distribute! Problem #19 CONT ( )( )= ( ) 4x2 4x 4x + 4 = 72x 72
4x 8x + 4 = 72x 72 2 2x 2x 2 2 4x2 80x + 76 = 0 4 4
4 Make equation equal to 0. Divide each term by 4. 4 x2 20x + 19 = 0 Problem #19 (By Factoring): x2 20x + 19 = 0 Solve: (x 19)(x 1) = 0
x = {1, 19} We should check our solutions! Problem #19 (By Quadratic Formula): x2 20x + 19 = 0 Use Quadratic Formula: ; Just plug in a, b, c Solve by quadratic formula! 1x2 20x + 19 = 0 So a = 1, b = ( 20), c = 19
Problem #19 CONT = = x = {10 9, 10 + 9} x = {1, 19} = We should check our solutions!
Problem #19 Checkingx = {1, 19} No Solution! 4 +5= 2 2 3 ? 4 ()+5= 2()2 3 9= 0 3 3 = -3 ? 4 ()+5= 2 () 2 3 81= 36 3 9 = 6 3 Problem #20 (Complex Numbers) Combine Like Terms! 36+7+2
=10 4 Problem #20 (Calculator Tip) = 2nd key + period (next to 0) =10 4 Problem #21 (Complex Numbers) Combine Like Terms! Distribute minus sign! (7 + 8) 1(9 + ) 7+8+9 1
=16 + 7 Problem #21 (Calculator Tip) = 2nd key + period (next to 0) =16 + 7 Problem #22 (Complex Numbers) Write the binomial twice! (8 + 9) (8 + 9) 64 + 72 2
+72 + 81 Multiply (FOIL)! Problem #22 CONT 64 + 72 2 +72 + 81 = 64 + 144 + 81(1) = 64 + 144 81 = 17 + 144 Remember: 2
= (1) Problem #22 (Calculator Tip) = 2nd key + period (next to 0) = 17 + 144 Problem #23 (Complex Numbers) Rationalize the denominator! To do so, multiply by the conjugate. Conjugates: (a + b)(a b) Conjugate of 8 + 2 82
(8 5 ) ( ) ( 8+2 ) ( ) Multiply/FOIL! Numerator: Problem #23 CONT Remember: 2 = (1) (8 5) (8 2) = 54 56 64 16 Numerator
2 40 + 10 = 64 56 + 10(1) = 64 56 10 Problem #23 CONTDenominator: (8 + 2) (8 2) 64 16 2 + 16 4 Remember: 2 = (1)
= 68 Denominator = 64 4(1) = 64 + 4 = 68 Problem #23 CONT Numerator = 54 56 Denominator = 68 54 56 = 68
= Problem #23 (Calculator Tip) = 2nd key + period (next to 0) Problem #23 Calculator CONT Change to Fraction: MATH KEY; Select 1) FRAC Problem #24 (Quadratic Equation) x2 4x 1) Half the middle
coefficient (b). 2) Square it! + = 13 3) Add it to both ______ + sides! (-2)2 = +4 x 4x + 4 = -9 2 Factor trinomial! Problem #24 CONT ; Match middle sign!
x 4x + 4 = -9 2 (x 2)(x 2) = -9 (x 2) = -9 = 2 x=2 x = {2 , 2 } x2= +
+ Problem #25 (Quadratic Equation) + + + + 2 x + 3x ______ = 9 9+ 1) Half the middle coefficient (b).
2) Square it! 3) Add it to both sides! ()2 = + Problem #25 CONT x + 3x + 2 (x + )(x + ) = (x + ) = 2 Factor trinomial! ;
Match middle sign! = Problem #25 CONT x+ = x+ = =3 Problem #25 CONT x+ =
x= = Problem #25 CONT + = ,
Problem #25 (Alternative Method) x + 3x 2 + + LCD = 4
1) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! + =9 + ______
()2 = + 4x2 + 12x + 9 = 36 + 9 4x2 + 12x + 9 = 45 Factor trinomial! Problem #25 CONT 4x + 12x + 9 = 45 2 ; Match middle sign!
(2x + 3)(2x + 3) = 45 2 (2x + 3) = 45 2x + 3 = 2x + 3 = = =3 Problem #25 CONT 2x + 3 =
2x = 3 2 x= 2 + = , Problem #26 (Quadratic Equation) 1) Half the middle
coefficient (b). 2) Square it! 3) Add it to both sides! 8x 5x = -1 2 8 8 8 Coefficient of
x2 must be 1! ()2 = + 2 x+ + Problem #26 CONT x2 x +
x2 x + Problem #26 CONT Factor trinomial! x x+ 2 ; Match middle sign! (x )(x ) = (x ) =
2 Problem #26 CONT (x ) = 2 = = = x = Problem #26 CONT
x = + x= + = Problem #26 CONT + = ,
Problem #26 (Alternative Method) 8x 5x = -1 2 8 8 Coefficient of x2 must be 1!
8 2 x+ + 1) Half the middle coefficient (b). 2) Square it! 3) Add it to both sides! ()2 = +
Problem #26 CONT 2 x LCD = 256
256x2 160x + 25 = 32 + 25 256x2 160x + 25 = 7 Factor trinomial! ; Match middle sign! (16x 5)(16x 5) = 7 (16x 5)2 = 7 Problem #26 CONT (16x 5) = 7 2 =
16x 5 = = 16x 5 = + + 16x = 5 16 16 x=
Problem #26 CONT x= + = , Problem #27 (Quadratic Equation) Use Quadratic Formula: ; Just plug in a, b, c Solve by quadratic formula! 1x2 + 10x + 3 = 0 So a = 1, b = 10, c = 3
Problem #27 CONT = = = All outsides numbers
are divisible by 2. Problem #27 CONT = x={} Problem #28 (Quadratic Equation) Make equation = 0 16x2 3x + 1 = 0 Use Quadratic Formula: ; Just plug in a, b, c
Solve by quadratic formula! 16x2 3x + 1 = 0 So a = 16, b = 3, c = 1 Solve by quadratic formula! Problem #28 CONT 16x2 3x + 1 = 0 So a = 16, b = 3, c = 1 = + =
, = Problem #29 (Square Root Method) (x + 7)2 = 24 Do opposite operations: Opposite of exponents Roots = x+7=
7 7 x= = x= Problem #30 (Quadratic Equation/ Problem Solving) Hitting the ground: Height = 0 -4.9t2 + 42t + 130 = 0
Solve by quadratic formula! -4.9t2 + 42t + 130 = 0 So a = (-4.9), b = 42, c = 130 Problem #30 CONT Use Quadratic Formula: ; Just plug in a, b, c Solve by quadratic formula! -4.9t2 + 42t + 130 = 0 So a = (-4.9), b = 42, c = 130 + = , .
. Problem #30 CONT + = , . . x {10.986, -2.4149} Extraneous Solution; Negative solution does not make sense! x 11.0 seconds Nearest tenth
Problem #31 (Quadratic Equation/ Problem Solving) Hitting the ground: Height = 0 -16t2 + 336t + 112 = 0 -16 -16 -16 1t2 21t 7 = 0
-16 Problem #31 CONT Use Quadratic Formula: ; Just plug in a, b, c Solve by quadratic formula! 1t2 21t 7 = 0 So a = 1, b = (-21), c = (-7) + = ,
Problem #31 CONT + = , x {-0.3282, 21.3282} Extraneous Solution; Negative solution does not make sense! x 21.3 seconds Nearest tenth Problem #32 (Vertex Problem)
Find vertex! Maximum value of parabola = Vertex (h, k). Vertex formula: t = (x-coordinate) h(t) = y-coordinate t= The arrow reaches the maximum height after 2 seconds. a = -16, b = 64 h(t) = -16t2 + 64t h(2) = -16(2)2 + 64(2) = 64 feet (Maximum height)
Problem #33 (Graphing Quadratics) f(x) = 1x2 + 2x 3 ; a = 1, b = 2 Vertex formula: x = (x-coordinate) f(x) = y-coordinate x= f(-1) = (-1)2 + 2(-1) 3 = -4 Since a = 1 (positive) Opens UPWARD! Vertex:
(-1, -4) Problem #33 (Sketch Graph): Vertex: (-1, -4) (-1, -4) Touches x-axis 2 times; 2 x-intercepts!! Problem #33 CONT y-intercept: x = 0 x-intercept: y = 0 x-intercept: y = 0 y-intercept: x = 0
f(0) = 02 + 2(0) 3 = 3 x2 + 2x 3 = 0 Solve by y-intercept: x2 + 2x 3 = 0 factoring! (0, 3) (x + 3)(x 1) = 0 x = {-3, 1} x-intercepts: (-3, 0), (1, 0) Problem #33 CONT Line of Symmetry: x = -1 (Vertical) x = h of vertex (h, k)
(-3, 0) (1, 0) (0, -3) Vertex: (-1, -4) y-intercept: (0, -3) x-intercepts: (-3, 0), (1, 0) Opens UPWARD! (-1, -4) Problem #33 Solution:
(1, 0) (-3, 0) (0, -3) (-1, -4) Domain and Range for all parabolas DOMAIN: (-, ) OR All real numbers k UP DOWN k RANGE: [k, )
RANGE: (-, k] Find Domain and Range for #33 y= Goes up forever x=- x= Goes to left forever! Goes to right forever! RANGE: [-4, )
DOMAIN: (-, ) OR All real numbers y = -4 (-1, -4) Range Visualized: (-1, -4) RANGE: [-4, ) y = -4 y= Goes up forever
Problem #34 (Graphing Quadratics) F(x) = 2x2 4x + 5; a = 2, b = 4 Vertex formula: x = (x-coordinate) f(x) = y-coordinate x= F(1) = 2(1)2 4(1) + 5 = 3 Since a = 2 (positive) Opens UPWARD!
Vertex: (1, 3) Problem #34 (Sketch Graph): Vertex: (1, 3) (1, 3) Does not touch x-axis; No x-intercepts!! Problem #34 CONT y-intercept: x = 0 x-intercept: y = 0 y-intercept: x = 0 F(0) = 2(0)2 4(0) + 5 = 5
y-intercept: (0, 5) x-intercept: y = 0 2x2 4x + 5 = 0 2x 4x + 5 = 0 2 Solve by quadratic formula! 2x2 4x + 5 = 0 So a = 2, b = (-4), c = 5 Problem #34 CONT Use Quadratic Formula:
; Just plug in a, b, c Quadratic Equation to solve: 2x2 4x + 5 = 0 So a = 2, b = (-4), c = 5 = NO SOLUTION! NO x-intercepts! = UNDEFINED! Problem #34 CONT (0, 5)
(2, 5) (1, 3) Line of Symmetry: Vertex: (1, 3) y-intercept: (0, 5) Another point: (2, 5) x-intercepts: NONE! Opens UPWARD! x = 1 (Vertical) x = h of vertex (h, k)
Problem #34 Solution: (0, 5) (1, 3) Find Domain and Range for #34 y= Goes up forever x= x=- Goes to right forever!
Goes to left forever! RANGE: [3, ) y=3 (1, 3) DOMAIN: (-, ) OR All real numbers Range Visualized: (1, 3) RANGE: [3, )
y=3 y= Goes up forever Problem #35 (Graphing Quadratics) Vertex form: f(x) = a(x h)2 + k Vertex = (h, k) x-coordinate: Opposite Sign y-coordinate: Copy! f(x) = 1(x + 2) 5 Vertex = (2, 5) Since a = 1 (positive) Opens
UPWARD! Vertex: (-2, -5) Line of Symmetry: 2 x = -2 (Vertical) x = h of vertex (h, k) Problem #35 CONT Vertex: (-2, -5) Opens UPWARD! Line of Symmetry:
x = -2 (Vertical) x = h of vertex (h, k) (-2, -5) Problem #35 Solution: (-2, -5) Find Domain and Range for #35 y= Goes up forever
x=- x= Goes to left forever! Goes to right forever! RANGE: [-5, ) y = -5 (-2, -5) DOMAIN: (-, ) OR All real numbers Range
Visualized: (-2, -5) RANGE: [-5, ) y = -5 y= Goes up forever Problem #36 (Graphing Quadratics) Vertex form: f(x) = a(x h)2 + k Vertex = (h, k)
x-coordinate: Opposite Sign y-coordinate: Copy! f(x) = -1(x 3) + 0 Vertex = (3, 0) Since a = -1 (negative) Opens DOWNWARDS! Vertex: (-2, -5) Line of Symmetry: 2 x = 3 (Vertical) x = h of vertex (h, k)
Problem #36 CONT Vertex: (3, 0) Opens DOWN! Line of Symmetry: x = 3 (Vertical) x = h of vertex (h, k) (3, 0) Problem #36
Solution (3, 0) Find Domain and Range for #36 y=0 (3, 0) RANGE: (, 0] x=- x= Goes to left forever! Goes to right forever!
y= Goes down forever DOMAIN: (-, ) OR All real numbers Range Visualized: y= RANGE: (, 0] (3, 0) Goes down forever
y=0 Problem #37 (Linear Graphing) y-intercept (x = 0): x-intercept (y = 0): x + 2y = 8 0 + 2y = 8 2y = 8 x + 2y = 8 x + 2(0) = 8 y=4
x=8 x y (0, 4) 0) (8, Problem #37 CONT (0, 4) (8, 0) Problem #37
Solution: Problem #38 (Linear Graphing) Make x-y table! x y (0, 3) (4, 6) x = 0: y= +3 x = 4: y= +3
y=3 y=6 Problem #38 CONT (4, 6) (0, 3) Solution for #38 (4, 6) (0, 3) Problem #38 (Alternative Method) y = mx + b (Slope-intercept form)
y-intercept = (0, b) Slope = m y= y-intercept: (0, 3) Slope = Up 3, Right 4 Problem #38 Alternative CONT y-intercept: (0, 3) Slope =
Up 3, Right 4 RIGHT 4 (4, 6) UP 3 (0, 3) Solution for #38 (4, 6) (0, 3) Find Domain and Range for #38 y=
Goes up forever x= x=- Goes to right forever! Goes to left forever! y= Goes down forever DOMAIN/RANGE: (-, ) OR All real numbers Problem #39 (Linear Graphing)
Find intercepts! y-intercept (x = 0): x-intercept (y = 0): -5x + 3y = -15 -50 + 3y = -15 3y = -15 -5x + 3y = -15 -5x + 3(0) = -15 -5x = -15 y = -5 x=3
x y (0, -5) (3, 0) Problem #39 CONT (3, 0) (0, -5) Solution for #39 (3, 0)
(0, -5) Problem #40 (Linear Graphing) Horizontal line; y = k Solution for #40 Problem #41 (Linear Graphing) Vertical line; x = k Solution for #41 Problem #42 (Slope-Intercept Form) Write the equation in the form y = mx + b
Point-slope form: y y1 = m(x x1) y ( 7) = -3(x 7) y + 7 = -3(x + 7) y +7 = -3x 21 7 7 y = -3x 28 Problem #43 (Function Notation) (, ) and (, ) (9, 43) and (1, 11)
Slope = Point-slope form: y y1 = m(x x1) Choose (1, 11) for (, ) and m = 4. Problem #43 CONT Choose (1, 11) for (, ) and m = 4. y 11 = 4(x 1) y 11 = 4x 4 +11 +11 y = 4x + 7 f(x) = 4x + 7
Problem #44 (Graphing Inequalities) Notes: y > mx + b Notes: <, >: Dotted Line Shade Above Line y < mx + b <, >: Solid Line Shade Below Line Graph: y < 2x + 6; Dotted Line/Below y > x 8; Solid Line/Above Problem #44 CONT
Graph: y < 2x + 6; Dotted Line/Below y > x 8; Solid Line/Above (1, 8) For y < 2x + 6: y-intercept = (0, 6) Slope = 2 (Go up 2, over by 1) (0, 6) For y > 1x 8: y-intercept = (0, -8)
Slope = 1 (Go up 1, over by 1) (1, -7) (0, -8) Problem #44 FINAL SOLUTION (1, 8) (0, 6) (1, -7) (0, -8) Problem #44 Solution Problem #45 (Systems of Equations) Let x = number of adult tickets
Let y = number of senior citizens tickets Solve for y (Substitution): Set-Up: x + y = 491 x + y = 491 tickets $25x + $13y = $8195 y = (491 x) Solve: 25x + 13(491 x) = 8195 25x + 6383 13x = 8195 Problem #45 CONT 25x + 6383 13x = 8195 12x + 6383 = 8195
12x = 1812 12 12 x = 151 adult tickets y = 491 x y = 491 151 = 340 senior citizen tickets Problem #46 (Systems of Equations)
Let x = pounds of trail mix Let y = pounds of cashew Set-Up: Solve (elimination): x + y = 75 pounds x + y = 75 $5x + $15y = ($13)(75) 5x + 15y = 975 Problem #46 CONT. We will eliminate x -5 (x + y = 75 ) 5x + 15y = 975
x + y = 75 x + 60 = 75 x = 15 pounds of trail mix -5x 5y = -375 5x + 15y = 975 10y = 600 10 10
y = 60 pounds of cashew Problem #47 (Rational Equations) Factor: x2 16 (x + 4)(x 4) 1 7 4 = ( +) ( ) ( +) ( ) Restrictions: Denominator 0 Solve:
x+4 0 Solve: x4 0 x -4 x4 Drop all solutions where x = {-4, 4} Problem #47 CONT LCD = (x + 4)(x 4)
1( +)( )7 ( +)( ) 4 ( +)( ) = ( +) ( ) ( +) ( ) 1(x 4) 7(x + 4) = 4 1x 4 7x 28 = 4 6x 32 = 4 + + 6x = 36 -6 -6
x=6 No extraneous solutions! Problem #48 (Rational Equations) Factor: x2 xx = Restrictions: Denominator 0 Drop all solutions where x = {0} x 0 2 LCD
= x (xx) Problem #48 CONT = x + 1x = 12 Make equation = 0 2 Factor trinomial!
x + 1x 12 = 0 x = {-4, 3} (x + 4)(x 3)= 0 No extraneous solutions! 2 Problem #49 (Rational Equations) Factor: x2 + 3x 4 (x + 4)(x 1) x2 2x + 1 (x 1)(x 1) ( + 5) ( 5) 5 = ( +)( ) ( ) ( ) ( +)( )
Restrictions: Denominator 0 Solve: x+4 0 Solve: x1 0 x -4 x1 Drop all solutions where x = {-4, 1}
LCD = (x + 4)(x 1)(x 1) Problem #49 CONT ( + 5) ( 5) 5 = ( +)( ) ( ) ( ) ( +)( ) ( +5 )( ) 5 ( + ) =( 5)( ) FOIL! DISTRIBUTE!
x2 1x + 5x 5 5x 20 FOIL! = x2 1x 5x + 5 x2 1x + 5x 5 5x 20 = x2 1x 5x + 5 x2 1x 25 = x2 6x + 5 Problem #49 CONT x 1x 25 = x 6x + 5 2 2
1x 25 = 6x + 5 + + 5x 25 = 5 + + 5x = 30 5
5 x=6 No extraneous solutions! Problem #50 (Compound Inequalities) Intersection ONLY! [-2, 3]
x < 3 (Shade left) x > -2 (Shade right) [ [ ] ] Problem #51 (Compound Inequalities) 11 < x + 6 < 31 22 < 5x + 12 < 62
10 < 5x < 50 5 5 5 2 < x < 10 x is between
2 and 10! Problem #51 (Compound Inequalities) 2 < x < 10; x is between 2 and 10! [ [2, 10) ) Remember: <, >: [] <, >: ()
Problem #52 (Compound Inequalities) x+4<1 x < -3 -4x < 4 -4 -4 x < -3 and x > -1 Intersection
ONLY! x > -1 Sign Flipped! ) ( There is no intersection/overlap! NO SOLUTION! Problem #52 Solution
Blank Number Line Problem #53 (Compound Inequalities) (, 4] ] [ (, 4] [6, )
Problem #54 (Compound Inequalities) ) ) Beginning (left) to End (right) ) (, 7)
Problem #55 (Compound Inequalities) -5x + 1 > 11 1 1 -5x > 10 -5 -5 x < -2 Sign Flipped!
3x + 3 > -9 3 3 3x > -12 3 3 x > -4 x < -2 or x > -4 Beginning to end! Problem #55 CONT
x < -2 or x > -4 Beginning to end! [ ] Beginning (left) to End (right) ( (, )
All real numbers! ) Problem #55 CONT ( (, ) All real numbers!
) Other Topics to Study: Functions, Domain/Range Midpoint and Distance Formula Problems Pythagorean Theorem Work Word Problems Review Factoring! Review Basic Graphing! Review Solving Equations and Simplifying Expressions Helpful Study Tips: 1) The final is cumulative and you should study
the course materials beyond this workshop! 2) The final exam is made by your individual instructor. Use any study guide/tips provided by your instructor. 3) Study your lab project worksheets and lab assessment. 4) Study previous class exams and quizzes! 5) Of course, study and review your homework assignments! Helpful Study Tips: 6) Visit the Math Connections for additional support and resources! Study a little each day, DO NOT CRAM!! General Test Taking Tips:
1) Preview the exam and do the problems that are easy and you are familiar with. 2) Pace yourself do not spend too much time on any 1 problem. 3) DO NOT RUSH! 4) Go back and check your answers (if time allows). 5) Follow instructions carefully! 6) Double check your work! Review your exam before you submit it! Now go study and do well on your final exam!
