mat 1033c final exam workshop - valencia college...table of contents • useful formulas •...
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MAT 1033C Final Exam Review
BY:Math Connections/Hands-On Math
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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
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Helpful Formulas to Remember…
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Slope/Linear Equations
m = 𝒚𝒚𝟐𝟐 − 𝒚𝒚𝟏𝟏𝒙𝒙𝟐𝟐 − 𝒙𝒙𝟏𝟏
Slope-intercept form: y = mx + bFunction Notation: f(x) = mx + bPoint-slope form: y – y1 = m(x – x1)Parallel Lines: Same SlopePerpendicular Lines: opposite, reciprocal slopes
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Work Formulas
Set-Up: 1𝒂𝒂
+ 1𝒃𝒃
+ 1𝒄𝒄
= 1𝒙𝒙
(3 people)
x = 𝒂𝒂𝒃𝒃𝒄𝒄𝒂𝒂𝒃𝒃+𝒂𝒂𝒄𝒄+𝒃𝒃𝒄𝒄
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Formula:
Factor: a3 + b3 OR a3 – b3
Formula: (a b)(a2 ab b2)
MMatch
OOpposite
PPlus
a and b are cube roots!
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3 𝑥𝑥3 = 𝒙𝒙 a3 8 = 2 b
(x 2)(x�x 2x 2�2)MMatch
–OOpposite
+PPlus
+
(x – 2)(x2 + 2x + 4)
(a b)(a�a ab b�b)
x3 – 8
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Problem #1 (Rational Expressions)
Factor & Cancel like factors!
9x4 – 72x 9x(x3 – 8) 9x(x – 2)(x2 + 2x + 4)
3x2 – 12 3(x2 – 4) 3(x – 2)(x + 2)x2 + 1x – 2 (x + 2)(x – 1)
4x3 + 8x2 + 16x 4x(x2 + 2x + 4)
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Problem #1 CONT…
9𝒙𝒙(𝒙𝒙 − 𝟐𝟐)(𝒙𝒙𝟐𝟐 + 𝟐𝟐𝒙𝒙 + 𝟒𝟒)3(𝒙𝒙 − 𝟐𝟐)(𝒙𝒙 + 𝟐𝟐)
�(𝒙𝒙 + 𝟐𝟐)(𝑥𝑥 − 1)
4𝒙𝒙(𝒙𝒙𝟐𝟐 + 𝟐𝟐𝒙𝒙 + 𝟒𝟒)
=9(𝒙𝒙 − 𝟏𝟏)
3 � 43
=3(𝒙𝒙 − 𝟏𝟏)
4
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Problem #2 (Rational Expressions)
Factor & Cancel like factors!
x2 + 13x + 36 (x + 9)(x + 4)x2 + 14x + 45 (x + 9)(x + 5)
x2 + 5x x(x + 5)x2 – 3x – 28 (x – 7)(x + 4)
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Problem #2 CONT…
(𝒙𝒙 + 𝟗𝟗)(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 + 𝟗𝟗)(𝒙𝒙 + 𝟓𝟓)
�𝒙𝒙(𝒙𝒙 + 𝟓𝟓)
(𝒙𝒙 − 𝟕𝟕)(𝒙𝒙 + 𝟒𝟒)
=𝒙𝒙
(𝒙𝒙 − 𝟕𝟕) =𝒙𝒙
𝒙𝒙 − 𝟕𝟕
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Problem #3 (Rational Expressions)
Division – Multiply by the reciprocal.“Keep, Change, Flip!”
Always flip the 2nd fraction!
𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙 − 𝟔𝟔𝒙𝒙𝟐𝟐 + 𝟗𝟗𝒙𝒙 + 𝟏𝟏𝟏𝟏
�𝒙𝒙𝟐𝟐 + 𝟕𝟕𝒙𝒙 + 𝟏𝟏𝟐𝟐
𝒙𝒙𝟐𝟐 − 𝟏𝟏
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𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙 − 𝟔𝟔𝒙𝒙𝟐𝟐 + 𝟗𝟗𝒙𝒙 + 𝟏𝟏𝟏𝟏
�𝒙𝒙𝟐𝟐 + 𝟕𝟕𝒙𝒙 + 𝟏𝟏𝟐𝟐
𝒙𝒙𝟐𝟐 − 𝟏𝟏
Problem #3 CONT…Factor &
Cancel like factors!
x2 + 5x – 6 (x + 6)(x – 1)x2 + 9x + 18 (x + 6)(x + 3)
x2 + 7x + 12 (x + 4)(x + 3)x2 – 1 (x – 1)(x + 1)
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Problem #3 CONT…
(𝒙𝒙 + 𝟔𝟔)(𝒙𝒙 − 𝟏𝟏)(𝒙𝒙 + 𝟔𝟔)(𝒙𝒙 + 𝟑𝟑)
�(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 + 𝟑𝟑)(𝒙𝒙 + 𝟏𝟏)(𝒙𝒙 − 𝟏𝟏)
=(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 + 𝟏𝟏) =
𝒙𝒙 + 𝟒𝟒𝒙𝒙 + 1
𝒙𝒙𝟐𝟐 + 𝟓𝟓𝒙𝒙 − 𝟔𝟔𝒙𝒙𝟐𝟐 + 𝟗𝟗𝒙𝒙 + 𝟏𝟏𝟏𝟏
�𝒙𝒙𝟐𝟐 + 𝟕𝟕𝒙𝒙 + 𝟏𝟏𝟐𝟐
𝒙𝒙𝟐𝟐 − 𝟏𝟏
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Problem #4 (Complex Fractions)Find LCD and multiply
each term by LCD! LCD = 12x
9 + 3𝑥𝑥
𝑥𝑥4 + 1
12
�12x �12x
�12x �12x3
108𝑥𝑥 + 363𝑥𝑥2 + 𝑥𝑥
=36(𝟑𝟑𝒙𝒙+𝟏𝟏)𝑥𝑥(𝟑𝟑𝒙𝒙+𝟏𝟏)
=𝟑𝟑𝟔𝟔𝒙𝒙
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Problem #5 (Complex Fractions)Find LCD and multiply
each term by LCD! LCD = x2
5𝑥𝑥 + 4
𝑥𝑥225𝑥𝑥2 − 16
𝑥𝑥
�x2 �x2
�x2 �x2
x
x
=𝟓𝟓𝒙𝒙 + 𝟒𝟒
𝟐𝟐𝟓𝟓 − 𝟏𝟏𝟔𝟔𝒙𝒙
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Problem #6 (Complex Fractions)Find LCD and multiply
each term by LCD! LCD = x2
1𝑥𝑥 + 9
𝑥𝑥2
𝑥𝑥 + 729𝑥𝑥2
�x2 �x2
�x2 �x2
x
𝑥𝑥 + 9𝒙𝒙𝟑𝟑 + 𝟕𝟕𝟐𝟐𝟗𝟗 Sum of Cubes Factor!
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Formula:
Factor: a3 + b3 OR a3 – b3
Formula: (a b)(a2 ab b2)
MMatch
OOpposite
PPlus
a and b are cube roots!
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3 𝑥𝑥3 = 𝒙𝒙 a3 729 = 9 b
(x 9)(x�x 9x 9�9)MMatch
+OOpposite
–PPlus
+
(x + 9)(x2 – 9x + 81)
(a b)(a�a ab b�b)
x3 + 729
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Problem #6 CONT…
𝑥𝑥 + 9𝒙𝒙𝟑𝟑 + 𝟕𝟕𝟐𝟐𝟗𝟗
= 𝟏𝟏(𝒙𝒙+𝟗𝟗)(𝒙𝒙+𝟗𝟗)(𝒙𝒙𝟐𝟐 −𝟗𝟗𝒙𝒙 + 𝟏𝟏𝟏𝟏)
=𝟏𝟏
𝒙𝒙𝟐𝟐 − 𝟗𝟗𝒙𝒙 + 𝟏𝟏𝟏𝟏
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Problem #7 (Complex Fractions)Find LCD and multiply
each term by LCD!
NOTE: 11 – x = – (x – 11)
−10𝒙𝒙 − 𝟏𝟏𝟏𝟏 + 11
𝑥𝑥 − 113𝑥𝑥 + 8
𝑥𝑥 − 11
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Problem #7 CONT…−10
(𝒙𝒙 − 𝟏𝟏𝟏𝟏) + 11(𝒙𝒙 − 𝟏𝟏𝟏𝟏)
3𝒙𝒙 + 8
(𝒙𝒙 − 𝟏𝟏𝟏𝟏)
LCD = x(x – 11)
�x(x – 11) �x(x – 11)
�x(x – 11) �x(x – 11)
= −10𝑥𝑥 + 11𝑥𝑥𝟑𝟑 𝒙𝒙 −𝟏𝟏𝟏𝟏 + 8𝑥𝑥
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Problem #7 CONT…
= −10𝑥𝑥 + 11𝑥𝑥𝟑𝟑 𝒙𝒙 −𝟏𝟏𝟏𝟏 + 8𝑥𝑥 = 𝑥𝑥
𝟑𝟑𝒙𝒙 −𝟑𝟑𝟑𝟑 +𝟏𝟏𝒙𝒙
=𝒙𝒙
𝟏𝟏𝟏𝟏𝒙𝒙 − 𝟑𝟑𝟑𝟑
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Problem #8 (Polynomial Division)
Steps on next slide!
(x – 8) 4x2 – 33x + 8
Long Division of polynomials!
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(x – 8) 4x2 – 33x + 8
x� ____ = 4x24x4x
4x(x – 8) = 4x2 – 32xChange signs!– 4x2 + 32x
– 1x + 8 x� ____ = -1x-1
-1(x – 8) = -1x + 8Change signs!
+ 1x – 8Remainder = 0
-1
4x - 1
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Problem #9 (Polynomial Division)
Steps on next slide!Long Division of polynomials!(3x + 5) 15x3 + 31x2 – 2x – 17
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(3x + 5) 15x3 + 31x2 – 2x – 173x� ____ = 15x35x25x2
5x2(3x + 5) = 15x3 + 25x2
Change signs!– 15x3 – 25x2
6x2 – 2x – 173x� ____ = 6x2+2x
+2x
2x(3x + 5) = 6x2 + 10xChange signs!
– 6x2 – 10x– 12x – 17
3x� ____ = -12x-4
– 4
-4(3x + 5) = -12x – 20Change signs!
+ 12x + 20Remainder = +3
5x2 + 2x – 4 + 𝟑𝟑(𝟑𝟑𝒙𝒙+𝟓𝟓)
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Problem #10 (Rational Exponents)
Find LCD:LCD for 3, 5, and 2 = 30
𝑥𝑥43 � 𝑥𝑥
65
𝑥𝑥−12
�𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 �
𝟔𝟔𝟔𝟔
�𝟏𝟏𝟓𝟓𝟏𝟏𝟓𝟓
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Problem #10 CONT…
Properties to Remember:1) xm � xn = xm + n
2) 𝐱𝐱𝐦𝐦
𝐱𝐱𝐧𝐧= 𝐱𝐱𝐦𝐦− 𝐧𝐧
𝑥𝑥40𝟑𝟑𝟏𝟏 � 𝑥𝑥
36𝟑𝟑𝟏𝟏
𝑥𝑥−15𝟑𝟑𝟏𝟏
𝟒𝟒𝟏𝟏𝟑𝟑𝟏𝟏
+𝟑𝟑𝟔𝟔𝟑𝟑𝟏𝟏
− (−𝟏𝟏𝟓𝟓𝟑𝟑𝟏𝟏
) =𝟗𝟗𝟏𝟏𝟑𝟑𝟏𝟏
= 𝑥𝑥𝟗𝟗𝟏𝟏𝟑𝟑𝟏𝟏
𝑥𝑥43 �
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 � 𝑥𝑥
65 �
𝟔𝟔𝟔𝟔
𝑥𝑥−12 �
𝟏𝟏𝟓𝟓𝟏𝟏𝟓𝟓
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Problem #11 (Rational Exponents)
= (𝟑𝟑𝑥𝑥𝟓𝟓𝟑𝟑)�(𝟑𝟑𝑥𝑥
𝟓𝟓𝟑𝟑)
𝑥𝑥16
Add exponents!xm � xn = xm + n
= 𝟗𝟗𝑥𝑥𝟏𝟏𝟏𝟏𝟑𝟑
𝑥𝑥𝟏𝟏𝟔𝟔
Subtract exponents!𝐱𝐱𝐦𝐦
𝐱𝐱𝐧𝐧= 𝐱𝐱𝐦𝐦− 𝐧𝐧
𝟏𝟏𝟏𝟏𝟑𝟑−𝟏𝟏𝟔𝟔
=206−
16
=𝟏𝟏𝟗𝟗𝟔𝟔
= 𝟗𝟗𝑥𝑥𝟏𝟏𝟗𝟗𝟔𝟔
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Problem #12 (Radicals)
𝟑𝟑𝟔𝟔𝒌𝒌𝟔𝟔𝒒𝒒𝟏𝟏 � 𝟐𝟐𝒌𝒌
6k3q4 𝟐𝟐𝒌𝒌
6k3q4 𝟐𝟐𝒌𝒌
𝒌𝒌𝟔𝟔 = 𝒌𝒌𝟑𝟑
𝒒𝒒𝟏𝟏 = 𝒒𝒒𝟒𝟒
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Problem #13 (Radicals)
= 𝟏𝟏𝟏𝟏𝟗𝟗𝑥𝑥5𝒚𝒚𝟔𝟔
𝟑𝟑𝒚𝒚𝟒𝟒
= 𝟔𝟔𝟑𝟑𝑥𝑥5𝒚𝒚𝟐𝟐
𝟗𝟗𝒙𝒙𝟒𝟒𝒚𝒚𝟐𝟐 � 𝟕𝟕𝒙𝒙𝟏𝟏
3x2y 𝟕𝟕𝒙𝒙
3x2y 𝟕𝟕𝒙𝒙
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Problem #14 (Radicals)
= 𝟗𝟗 + 𝟐𝟐𝟏𝟏 + 𝟑𝟑𝟔𝟔 + 𝟒𝟒𝟏𝟏𝟓𝟓= 𝟑𝟑 + 𝟒𝟒 𝟓𝟓 + 𝟔𝟔 + 𝟏𝟏𝟏𝟏 𝟓𝟓= 𝟑𝟑 + 𝟐𝟐 𝟓𝟓 + 𝟔𝟔 + 𝟗𝟗 𝟓𝟓
= 𝟗𝟗 + 𝟏𝟏𝟏𝟏 𝟓𝟓
NOTE:𝟗𝟗 = 3𝟒𝟒 = 2𝟑𝟑𝟔𝟔 = 6𝟏𝟏𝟏𝟏 = 9
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Problem #15 (Radicals)
= 𝟑𝟑 𝟐𝟐𝟕𝟕𝒚𝒚 − 𝟑𝟑 𝟏𝟏𝟐𝟐𝟏𝟏𝒚𝒚= 𝟑𝟑𝟑𝟑 𝒚𝒚 − 𝟑𝟑 𝟔𝟔𝟒𝟒 𝟑𝟑 𝟐𝟐𝒚𝒚
=𝟑𝟑𝟑𝟑 𝒚𝒚 − 𝟒𝟒𝟑𝟑 𝟐𝟐𝒚𝒚
= 𝟑𝟑𝟑𝟑 𝒚𝒚 − 𝟒𝟒𝟑𝟑 𝟐𝟐𝒚𝒚
NOTE:𝟑𝟑 𝟐𝟐𝟕𝟕 = 3Since 3�3�3 = 27𝟑𝟑 𝟔𝟔𝟒𝟒 = 4Since 4�4�4 = 64
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Problem #16 (Radicals)
2𝑥𝑥 − 1 + 𝟒𝟒 = 10– 4
Isolate radical!
– 4
2𝑥𝑥 − 1 = 𝟔𝟔2 2
2x – 1 = 𝟑𝟑𝟔𝟔+ 1 + 1
2x = 𝟑𝟑𝟕𝟕
x = 𝟑𝟑𝟕𝟕𝟐𝟐
2 2
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Problem #16 - - Checking:
2𝒙𝒙 − 1 + 4 = 10 x = 𝟑𝟑𝟕𝟕𝟐𝟐
2 �𝟑𝟑𝟕𝟕𝟐𝟐 − 1 + 4 = 10
?
𝟑𝟑𝟕𝟕 − 𝟏𝟏 + 4 = 10𝟑𝟑𝟔𝟔 + 4 = 10𝟔𝟔 + 4 = 10
?
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Problem #17 (Radicals)
31 − 𝑥𝑥 = (𝑥𝑥 − 1)
Square both sides!
2 2
𝟑𝟑𝟏𝟏 − 𝒙𝒙 = (𝒙𝒙 − 𝟏𝟏)(𝒙𝒙 − 𝟏𝟏) FOIL!
𝟑𝟑𝟏𝟏 − 𝒙𝒙 = 𝒙𝒙𝟐𝟐 − 𝟐𝟐𝒙𝒙 + 𝟏𝟏 Make equationequal to zero!
– 31 – 31+ x + 1x
0 = 𝒙𝒙𝟐𝟐 − 𝒙𝒙 − 𝟑𝟑𝟏𝟏
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Problem #17 CONT…
𝒙𝒙𝟐𝟐 − 𝟏𝟏𝒙𝒙 − 𝟑𝟑𝟏𝟏 = 0
Factor trinomial!
(𝒙𝒙 )(𝒙𝒙 ) = 0–30
1 � 302 � 15
3 � 10
5 � 6
– 6 + 5
Solve:x – 6 = 0; x + 5 = 0
Solutions: x = 6; x = – 5+6 +6 – 5 – 5
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Problem #17 - - Checking!x = 6; x = – 5
31 − 𝒙𝒙 = 𝒙𝒙 − 1
31 − 𝟔𝟔 = 𝟔𝟔 − 1?
25 = 5
31 − (−𝟓𝟓) = −𝟓𝟓 − 1?
36 = −𝟔𝟔
x = 6
Extraneous Solution
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Problem #18 (Radicals) Square both sides!
2𝑥𝑥 + 5 = (3 + 𝑥𝑥 − 2)2 2
2𝑥𝑥 + 5 = (3 + 𝑥𝑥 − 2) (3 + 𝑥𝑥 − 2)FOIL!
2𝑥𝑥 + 5 = 9+𝟑𝟑 𝒙𝒙 − 𝟐𝟐
+𝟑𝟑 𝒙𝒙 − 𝟐𝟐 + 𝒙𝒙 − 𝟐𝟐𝟐𝟐
2𝑥𝑥 + 5 = 9 + 𝟔𝟔 𝒙𝒙 − 𝟐𝟐 + x – 2
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Problem #18 CONT…2𝑥𝑥 + 5 = 9 + 𝟔𝟔 𝒙𝒙 − 𝟐𝟐 + x – 2
2𝑥𝑥 + 5 = 7 + 𝟔𝟔 𝒙𝒙 − 𝟐𝟐 + x Isolate 𝟔𝟔 𝒙𝒙 − 𝟐𝟐– 7 – 7
𝑥𝑥 − 2 = [𝟔𝟔 𝒙𝒙 − 𝟐𝟐]2 2
𝑥𝑥 − 2 𝑥𝑥 − 2 = 𝟑𝟑𝟔𝟔(𝒙𝒙 − 𝟐𝟐)FOIL!
x2 – 4x + 4 = 36x – 72Distribute!
– x – x
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Problem #18 CONT….
x2 – 4x + 4 = 36x – 72 Make equationequal to zero!– 36x – 36x+72 +72
x2 – 40x + 76 = 0(x – 38)(x – 2) = 0
Solve by factoring!
+76
1 � 762 � 384 � 19
x = {2, 38}Check your solutions!
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Problem #18 - - Checking!x = 2; x = 38
2𝒙𝒙 + 5 = 3 + 𝒙𝒙 − 22 � 𝟐𝟐 + 5 = 3 + 𝟐𝟐 − 2
9 = 3 + 02 � 𝟑𝟑𝟏𝟏 + 5 = 3 + 𝟑𝟑𝟏𝟏 − 2
81 = 3 + 36
NOExtraneous Solutions!
3 = 3
9 = 3 + 6
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Problem #19 (Radicals)
4𝑥𝑥 + 5 = ( 2𝑥𝑥 − 2 − 3)
Square both sides!
2 2
𝟒𝟒𝒙𝒙 + 𝟓𝟓 = ( 𝟐𝟐𝒙𝒙 − 𝟐𝟐 − 3)( 𝟐𝟐𝒙𝒙 − 𝟐𝟐 − 3)FOIL!
𝟒𝟒𝒙𝒙 + 𝟓𝟓 = 𝟐𝟐𝒙𝒙 − 𝟐𝟐𝟐𝟐− 𝟑𝟑 𝟐𝟐𝒙𝒙 − 𝟐𝟐−𝟑𝟑 𝟐𝟐𝒙𝒙 − 𝟐𝟐 + 𝟗𝟗
𝟒𝟒𝒙𝒙 + 𝟓𝟓 = 𝟐𝟐𝒙𝒙 − 𝟐𝟐 − 𝟔𝟔 𝟐𝟐𝒙𝒙 − 𝟐𝟐 + 𝟗𝟗
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Problem #19 CONT…
𝟒𝟒𝒙𝒙 + 𝟓𝟓 = 𝟐𝟐𝒙𝒙 − 𝟐𝟐 − 𝟔𝟔 𝟐𝟐𝒙𝒙 − 𝟐𝟐 + 𝟗𝟗𝟒𝟒𝒙𝒙 + 𝟓𝟓 = 𝟐𝟐𝒙𝒙 + 7 − 𝟔𝟔 𝟐𝟐𝒙𝒙 − 𝟐𝟐
Isolate -6 𝟐𝟐𝒙𝒙 − 𝟐𝟐
−𝟐𝟐𝒙𝒙 −𝟐𝟐𝒙𝒙−𝟕𝟕 −𝟕𝟕
(𝟐𝟐𝒙𝒙 − 𝟐𝟐) = [−𝟔𝟔 𝟐𝟐𝒙𝒙 − 𝟐𝟐]Square Both Sides!
2 2
(𝟐𝟐𝒙𝒙 − 𝟐𝟐)(𝟐𝟐𝒙𝒙 − 𝟐𝟐) = 𝟑𝟑𝟔𝟔(𝟐𝟐𝒙𝒙 − 𝟐𝟐)FOIL! Distribute!
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Problem #19 CONT…
(𝟐𝟐𝒙𝒙 − 𝟐𝟐)(𝟐𝟐𝒙𝒙 − 𝟐𝟐) = 𝟑𝟑𝟔𝟔(𝟐𝟐𝒙𝒙 − 𝟐𝟐)
4x2 – 4x – 4x + 4 = 72x – 72
4x2 – 8x + 4 = 72x – 72 Make equationequal to 0.
−𝟕𝟕2x −𝟕𝟕2x+𝟕𝟕2 +𝟕𝟕2
4x2 – 80x + 76 = 0 Divide each term by 4.
4 4 4 4
x2 – 20x + 19 = 0
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Problem #19 (By Factoring):
Solve:(x – 19)(x – 1) = 0
x = {1, 19} We should checkour solutions!
x2 – 20x + 19 = 0
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Problem #19 (By Quadratic Formula):
x2 – 20x + 19 = 0Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
𝒙𝒙 = −(−𝟐𝟐𝟏𝟏)± (−𝟐𝟐𝟏𝟏)𝟐𝟐−𝟒𝟒(𝟏𝟏)(𝟏𝟏𝟗𝟗)𝟐𝟐(𝟏𝟏)
Solve by quadratic formula!1x2 – 20x + 19 = 0
So a = 1, b = (– 20), c = 19
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Problem #19 CONT…
𝒙𝒙 = −(−𝟐𝟐𝟏𝟏)± (−𝟐𝟐𝟏𝟏)𝟐𝟐−𝟒𝟒(𝟏𝟏)(𝟏𝟏𝟗𝟗)𝟐𝟐(𝟏𝟏)
𝒙𝒙 =𝟐𝟐𝟏𝟏 ± 𝟑𝟑𝟐𝟐𝟒𝟒
𝟐𝟐𝟑𝟑𝟐𝟐𝟒𝟒 = 𝟏𝟏𝟏𝟏
𝒙𝒙 =𝟐𝟐𝟏𝟏 ± 𝟏𝟏𝟏𝟏
𝟐𝟐 = 𝟐𝟐𝟏𝟏𝟐𝟐
± 𝟏𝟏𝟏𝟏𝟐𝟐
= 𝟏𝟏𝟏𝟏 ± 𝟗𝟗
x = {1, 19}x = {10 – 9, 10 + 9}
We should checkour solutions!
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Problem #19 Checking…x = {1, 19}
4𝒙𝒙 + 5 = 2𝒙𝒙 − 2 − 3
4(𝟏𝟏) + 5 = 2(𝟏𝟏) − 2 − 3?
9 = 0 − 3 3 = -3
4(𝟏𝟏𝟗𝟗) + 5 = 2(𝟏𝟏𝟗𝟗) − 2 − 3?
81 = 36 − 3 9 = 6 – 3
No Solution!∅
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Problem #20 (Complex Numbers)
Combine Like Terms!
3 – 6𝒊𝒊 + 7 + 2𝒊𝒊=10 – 4𝒊𝒊
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Problem #20 (Calculator Tip)
=10 – 4𝒊𝒊𝒊𝒊 = 2nd key + period (next to 0)
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Problem #21 (Complex Numbers)
Combine Like Terms!Distribute minus sign!
(7 + 8𝑖𝑖) –1(–9 + 𝑖𝑖)7 + 8𝒊𝒊 + 9 – 1𝒊𝒊
=16 + 7𝒊𝒊
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Problem #21 (Calculator Tip)
=16 + 7𝒊𝒊𝒊𝒊 = 2nd key + period (next to 0)
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Problem #22 (Complex Numbers)
Write the binomial twice!
(8 + 9𝑖𝑖) (8 + 9𝑖𝑖) Multiply (FOIL)!
64 + 72𝑖𝑖+72𝑖𝑖 + 81𝑖𝑖2
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Problem #22 CONT…
64 + 72𝑖𝑖+72𝑖𝑖 + 81𝑖𝑖2
= 64 + 144𝒊𝒊 + 81(–1)
Remember:𝒊𝒊2 = (–1)
= 64 + 144𝒊𝒊 – 81
= – 17 + 144𝒊𝒊
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Problem #22 (Calculator Tip)
𝒊𝒊 = 2nd key + period (next to 0)
= – 17 + 144𝒊𝒊
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Problem #23 (Complex Numbers)
Rationalize the denominator!To do so, multiply by the conjugate.Conjugates: (a + b)(a – b)
Conjugate of 8 + 2𝒊𝒊8 – 2𝒊𝒊
(8 − 5𝑖𝑖)(8 + 2𝑖𝑖)
�(𝟏𝟏 − 𝟐𝟐𝒊𝒊)(𝟏𝟏 − 𝟐𝟐𝒊𝒊)
Multiply/FOIL!
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Problem #23 CONT… Numerator:
(8 – 5𝑖𝑖) (8 – 2𝑖𝑖) 64 – 16𝑖𝑖
– 40𝑖𝑖 + 10𝒊𝒊2
Remember:𝒊𝒊2 = (–1)
= 64 – 56𝒊𝒊 + 10(–1)= 64 – 56𝒊𝒊 – 10
= 54 – 56𝒊𝒊Numerator
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Problem #23 CONT…Denominator:
(8 + 2𝑖𝑖) (8 – 2𝑖𝑖) 64 – 16𝑖𝑖
+ 16𝑖𝑖 – 4𝒊𝒊2
Remember:𝒊𝒊2 = (–1)
= 64 – 4(–1) = 64 + 4 = 68
= 68Denominator
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Problem #23 CONT…
Numerator = 54 – 56𝒊𝒊Denominator = 68
54 − 56𝑖𝑖68
= 5468
− 5668
𝑖𝑖
= 2734− 14
17𝑖𝑖
÷ 𝟐𝟐
÷ 𝟐𝟐
÷ 𝟒𝟒
÷ 𝟒𝟒
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Problem #23 (Calculator Tip)
𝒊𝒊 = 2nd key + period (next to 0)
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Problem #23 Calculator CONT…
Change to Fraction: MATH KEY; Select 1) FRAC
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Problem #24 (Quadratic Equation)
−𝟏𝟏𝟑𝟑 −𝟏𝟏𝟑𝟑
x2 – 4x ______ = –13
1) Half the middle coefficient (b).2) Square it!3) Add it to both sides!
𝟏𝟏𝟐𝟐−𝟒𝟒 = −𝟐𝟐;(-2)2 = +4
+𝟒𝟒
+𝟒𝟒
x2 – 4x + 4 = -9 Factor trinomial!
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Problem #24 CONT…
x2 – 4x + 4 = -9 𝒙𝒙𝟐𝟐 = 𝒙𝒙; 𝟒𝟒 = 𝟐𝟐Match middle sign!
(x – 2)(x – 2) = -9
(x – 2)2 = -9x – 2 = ± −𝟗𝟗
+𝟐𝟐 +𝟐𝟐x = 2 ±𝟑𝟑𝒊𝒊x = {2 −𝟑𝟑𝒊𝒊, 2 +𝟑𝟑𝒊𝒊}
(𝒙𝒙 − 𝟐𝟐)𝟐𝟐 = ± −𝟗𝟗
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Problem #25 (Quadratic Equation)
+𝟗𝟗 +𝟗𝟗
x2 + 3x ______ = 9
1) Half the middle coefficient (b).2) Square it!3) Add it to both sides!
𝟏𝟏𝟐𝟐𝟑𝟑 =
𝟑𝟑𝟐𝟐
;
(𝟑𝟑𝟐𝟐)2 = +𝟗𝟗
𝟒𝟒
+𝟗𝟗𝟒𝟒
+𝟗𝟗𝟒𝟒
9 + 𝟗𝟗𝟒𝟒
= 𝟗𝟗𝟏𝟏
+ 𝟗𝟗𝟒𝟒
= 𝟑𝟑𝟔𝟔𝟒𝟒
+ 𝟗𝟗𝟒𝟒
= 𝟒𝟒𝟓𝟓𝟒𝟒
� 𝟒𝟒
� 𝟒𝟒
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Problem #25 CONT…
x2 + 3x + 𝟗𝟗𝟒𝟒
= 𝟒𝟒𝟓𝟓𝟒𝟒
Factor trinomial!
𝒙𝒙𝟐𝟐 = 𝒙𝒙; 𝟗𝟗𝟒𝟒
= 𝟑𝟑𝟐𝟐
Match middle sign!(x + 𝟑𝟑
𝟐𝟐)(x + 𝟑𝟑
𝟐𝟐) = 𝟒𝟒𝟓𝟓
𝟒𝟒
(x + 𝟑𝟑𝟐𝟐)2 = 𝟒𝟒𝟓𝟓
𝟒𝟒(𝒙𝒙 + 𝟑𝟑
𝟐𝟐)𝟐𝟐 = ± 𝟒𝟒𝟓𝟓
𝟒𝟒
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Problem #25 CONT…
x + 𝟑𝟑𝟐𝟐
= ± 𝟒𝟒𝟓𝟓𝟒𝟒
𝟒𝟒𝟓𝟓 = 𝟗𝟗 � 𝟓𝟓=3 𝟓𝟓
x + 𝟑𝟑𝟐𝟐
= ± 𝟑𝟑 𝟓𝟓𝟐𝟐
𝟒𝟒 = 𝟐𝟐
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Problem #25 CONT…
x + 𝟑𝟑𝟐𝟐
= ± 𝟑𝟑 𝟓𝟓𝟐𝟐
−𝟑𝟑𝟐𝟐 −
𝟑𝟑𝟐𝟐
x =−𝟑𝟑𝟐𝟐
± 𝟑𝟑 𝟓𝟓𝟐𝟐
= −𝟑𝟑 ± 𝟑𝟑 𝟓𝟓𝟐𝟐
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Problem #25 CONT…
−𝟑𝟑 ± 𝟑𝟑 𝟓𝟓𝟐𝟐
𝒙𝒙 =−𝟑𝟑 − 𝟑𝟑 𝟓𝟓
𝟐𝟐,−𝟑𝟑 + 𝟑𝟑 𝟓𝟓
𝟐𝟐
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Problem #25 (Alternative Method)
+𝟗𝟗 +𝟗𝟗
x2 + 3x ______ = 9
1) Half the middle coefficient (b).2) Square it!3) Add it to both sides!
𝟏𝟏𝟐𝟐𝟑𝟑 =
𝟑𝟑𝟐𝟐
;
(𝟑𝟑𝟐𝟐)2 = +𝟗𝟗
𝟒𝟒
+𝟗𝟗𝟒𝟒
+𝟗𝟗𝟒𝟒
𝟒𝟒 � 𝟒𝟒 �𝟒𝟒 �
𝟒𝟒 �� 𝟒𝟒
4x2 + 12x + 9 = 36 + 9
4x2 + 12x + 9 = 45 Factor trinomial!
LCD = 4
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Problem #25 CONT…𝟒𝟒𝒙𝒙𝟐𝟐 = 𝟐𝟐𝒙𝒙; 𝟗𝟗 = 𝟑𝟑
Match middle sign!
(2x + 3)(2x + 3) = 45(2x + 3)2 = 45
2x + 3 = ± 𝟒𝟒𝟓𝟓
4x2 + 12x + 9 = 45
𝟒𝟒𝟓𝟓 = 𝟗𝟗 � 𝟓𝟓=3 𝟓𝟓
2x + 3 = ±𝟑𝟑 𝟓𝟓
(𝟐𝟐𝒙𝒙 + 𝟑𝟑)𝟐𝟐 = ± 𝟒𝟒𝟓𝟓
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Problem #25 CONT…
2x + 3 = ±𝟑𝟑 𝟓𝟓−𝟑𝟑 −𝟑𝟑
2x = –3 ±𝟑𝟑 𝟓𝟓2 2
x = −𝟑𝟑±𝟑𝟑 𝟓𝟓𝟐𝟐
𝒙𝒙 =−𝟑𝟑 − 𝟑𝟑 𝟓𝟓
𝟐𝟐,−𝟑𝟑 + 𝟑𝟑 𝟓𝟓
𝟐𝟐
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Problem #26 (Quadratic Equation)
−𝟏𝟏 −𝟏𝟏
8x2 – 5x = -18
Coefficient of x2 must be 1!
8 8
x2 −𝟓𝟓𝟏𝟏𝒙𝒙 = −𝟏𝟏
𝟏𝟏
1) Half the middle coefficient (b).2) Square it!3) Add it to both sides!
𝟏𝟏𝟐𝟐
−𝟓𝟓𝟏𝟏
= −𝟓𝟓𝟏𝟏𝟔𝟔
;
(− 𝟓𝟓𝟏𝟏𝟔𝟔
)2 = + 𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
+𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
+𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
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Problem #26 CONT…
−𝟏𝟏𝟏𝟏
+ 𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
= − 𝟑𝟑𝟐𝟐𝟐𝟐𝟓𝟓𝟔𝟔
+ 𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
= − 𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔� 𝟑𝟑𝟐𝟐
x2 – 𝟓𝟓𝟏𝟏x + 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔= −𝟏𝟏
𝟏𝟏+ 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔
� 𝟑𝟑𝟐𝟐
x2 – 𝟓𝟓𝟏𝟏x + 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔= − 𝟕𝟕
𝟐𝟐𝟓𝟓𝟔𝟔
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Problem #26 CONT…
x2 – 𝟓𝟓𝟏𝟏x + 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔= − 𝟕𝟕
𝟐𝟐𝟓𝟓𝟔𝟔Factor trinomial!
𝒙𝒙𝟐𝟐 = 𝒙𝒙; 𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
= 𝟓𝟓𝟏𝟏𝟔𝟔
Match middle sign!
(x – 𝟓𝟓𝟏𝟏𝟔𝟔
)(x – 𝟓𝟓𝟏𝟏𝟔𝟔
) = − 𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔
(x – 𝟓𝟓𝟏𝟏𝟔𝟔
)2 = − 𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔
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Problem #26 CONT…
(x – 𝟓𝟓𝟏𝟏𝟔𝟔
)2 = − 𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔
(𝒙𝒙 − 𝟓𝟓𝟏𝟏𝟔𝟔
)𝟐𝟐 = ± −𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔
−𝟕𝟕 = −1 � 7= 𝒊𝒊 𝟕𝟕𝟐𝟐𝟓𝟓𝟔𝟔 = 𝟏𝟏𝟔𝟔
x – 𝟓𝟓𝟏𝟏𝟔𝟔
= ± 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
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Problem #26 CONT…
x – 𝟓𝟓𝟏𝟏𝟔𝟔
= ± 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
+𝟓𝟓𝟏𝟏𝟔𝟔 +
𝟓𝟓𝟏𝟏𝟔𝟔
x = 𝟓𝟓𝟏𝟏𝟔𝟔
± 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
= 𝟓𝟓 ± 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
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Problem #26 CONT…
𝟓𝟓 ± 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
𝒙𝒙 =𝟓𝟓 − 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
,𝟓𝟓 + 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
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Problem #26 (Alternative Method)
−𝟏𝟏 −𝟏𝟏
8x2 – 5x = -18
Coefficient of x2 must be 1!
8 8
x2 −𝟓𝟓𝟏𝟏𝒙𝒙 = −𝟏𝟏
𝟏𝟏
1) Half the middle coefficient (b).2) Square it!3) Add it to both sides!
𝟏𝟏𝟐𝟐
−𝟓𝟓𝟏𝟏
= −𝟓𝟓𝟏𝟏𝟔𝟔
;
(− 𝟓𝟓𝟏𝟏𝟔𝟔
)2 = + 𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
+𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
+𝟐𝟐𝟓𝟓𝟐𝟐𝟓𝟓𝟔𝟔
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Problem #26 CONT…
x2 −𝟓𝟓𝟏𝟏𝒙𝒙 + 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔= −𝟏𝟏
𝟏𝟏+ 𝟐𝟐𝟓𝟓
𝟐𝟐𝟓𝟓𝟔𝟔
� 𝟐𝟐𝟓𝟓𝟔𝟔
� 𝟐𝟐𝟓𝟓𝟔𝟔 � 𝟐𝟐𝟓𝟓𝟔𝟔 � 𝟐𝟐𝟓𝟓𝟔𝟔 � 𝟐𝟐𝟓𝟓𝟔𝟔
LCD = 256� 𝟑𝟑𝟐𝟐 � 𝟑𝟑𝟐𝟐
256x2 – 160x + 25 = – 32 + 25
256x2 – 160x + 25 = – 7 Factor trinomial!
𝟐𝟐𝟓𝟓𝟔𝟔𝒙𝒙𝟐𝟐 = 𝟏𝟏𝟔𝟔𝒙𝒙; 𝟐𝟐𝟓𝟓 = 𝟓𝟓Match middle sign!(16x – 5)(16x – 5) = – 7
(16x – 5)2 = – 7
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Problem #26 CONT… (16x – 5)2 = – 7
16x – 5 = ± −𝟕𝟕 −𝟕𝟕 = −1 � 7= 𝒊𝒊 𝟕𝟕
16x – 5 = ±𝒊𝒊 𝟕𝟕+𝟓𝟓 +𝟓𝟓
16x = 5 ±𝒊𝒊 𝟕𝟕16 16
x = 𝟓𝟓±𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
(𝟏𝟏𝟔𝟔𝒙𝒙 − 𝟓𝟓)𝟐𝟐 = ± −𝟕𝟕
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Problem #26 CONT…
x = 𝟓𝟓±𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
𝒙𝒙 =𝟓𝟓 − 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
,𝟓𝟓 + 𝒊𝒊 𝟕𝟕𝟏𝟏𝟔𝟔
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Problem #27 (Quadratic Equation)
Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
𝒙𝒙 = −𝟏𝟏𝟏𝟏± 𝟏𝟏𝟏𝟏𝟐𝟐−𝟒𝟒(𝟏𝟏)(𝟑𝟑)𝟐𝟐(𝟏𝟏)
Solve by quadratic formula!1x2 + 10x + 3 = 0
So a = 1, b = 10, c = 3
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Problem #27 CONT…
𝒙𝒙 =−𝟏𝟏𝟏𝟏 ± 𝟏𝟏𝟏𝟏
𝟐𝟐
𝒙𝒙 = −𝟏𝟏𝟏𝟏± 𝟏𝟏𝟏𝟏𝟐𝟐−𝟒𝟒(𝟏𝟏)(𝟑𝟑)𝟐𝟐(𝟏𝟏)
𝟏𝟏𝟏𝟏 = 4 � 22= 𝟐𝟐 𝟐𝟐𝟐𝟐
𝒙𝒙 =−𝟏𝟏𝟏𝟏 ± 𝟐𝟐 𝟐𝟐𝟐𝟐
𝟐𝟐
All outsides numbersare divisible by 2.
÷ 𝟐𝟐 ÷ 𝟐𝟐
÷ 𝟐𝟐
𝒙𝒙 = −𝟓𝟓±𝟏𝟏 𝟐𝟐𝟐𝟐𝟏𝟏
= −𝟓𝟓 ± 𝟐𝟐𝟐𝟐
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Problem #27 CONT…
𝒙𝒙 = −𝟓𝟓 ± 𝟐𝟐𝟐𝟐x = {−𝟓𝟓 − 𝟐𝟐𝟐𝟐, −𝟓𝟓 + 𝟐𝟐𝟐𝟐}
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Problem #28 (Quadratic Equation)
Make equation = 0−𝟑𝟑𝒙𝒙−𝟑𝟑𝒙𝒙 16x2 – 3x + 1 = 0
Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
Solve by quadratic formula!16x2 – 3x + 1 = 0
So a = 16, b = – 3, c = 1
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Problem #28 CONT…
𝒙𝒙 = −(−𝟑𝟑)± (−𝟑𝟑)𝟐𝟐−𝟒𝟒(𝟏𝟏𝟔𝟔)(𝟏𝟏)𝟐𝟐(𝟏𝟏𝟔𝟔)
Solve by quadratic formula!16x2 – 3x + 1 = 0
So a = 16, b = – 3, c = 1
𝒙𝒙 =𝟑𝟑 ± −𝟓𝟓𝟓𝟓
𝟑𝟑𝟐𝟐
𝒙𝒙 =𝟑𝟑 − 𝒊𝒊 𝟓𝟓𝟓𝟓
𝟑𝟑𝟐𝟐,𝟑𝟑 + 𝒊𝒊 𝟓𝟓𝟓𝟓
𝟑𝟑𝟐𝟐
−𝟓𝟓𝟓𝟓 = −1 � 55= 𝒊𝒊 𝟓𝟓𝟓𝟓
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Problem #29 (Square Root Method)
Do opposite operations:Opposite of exponents Roots(x + 7)2 = 24
(𝒙𝒙 + 𝟕𝟕)𝟐𝟐 = ± 𝟐𝟐𝟒𝟒
x + 7 = ± 𝟐𝟐𝟒𝟒–7 –7
𝟐𝟐𝟒𝟒 = 4 � 6 = 𝟐𝟐 𝟔𝟔
x = −𝟕𝟕 ± 𝟐𝟐 𝟔𝟔x = {−𝟕𝟕 − 𝟐𝟐 𝟔𝟔,−𝟕𝟕 + 𝟐𝟐 𝟔𝟔}
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Problem #30 (Quadratic Equation/Problem Solving)
Hitting the ground:Height = 0
-4.9t2 + 42t + 130 = 0Solve by quadratic formula!
-4.9t2 + 42t + 130 = 0So a = (-4.9), b = 42, c = 130
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Problem #30 CONT…Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
𝒙𝒙 = −𝟒𝟒𝟐𝟐± 𝟒𝟒𝟐𝟐𝟐𝟐−𝟒𝟒(−𝟒𝟒.𝟗𝟗)(𝟏𝟏𝟑𝟑𝟏𝟏)𝟐𝟐(−𝟒𝟒.𝟗𝟗)
𝒙𝒙 =−𝟒𝟒𝟐𝟐 − 𝟒𝟒𝟑𝟑𝟏𝟏𝟐𝟐
−𝟗𝟗.𝟏𝟏,−𝟒𝟒𝟐𝟐 + 𝟒𝟒𝟑𝟑𝟏𝟏𝟐𝟐
−𝟗𝟗.𝟏𝟏
Solve by quadratic formula!-4.9t2 + 42t + 130 = 0
So a = (-4.9), b = 42, c = 130
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Problem #30 CONT…
𝒙𝒙 =−𝟒𝟒𝟐𝟐 − 𝟒𝟒𝟑𝟑𝟏𝟏𝟐𝟐
−𝟗𝟗.𝟏𝟏,−𝟒𝟒𝟐𝟐 + 𝟒𝟒𝟑𝟑𝟏𝟏𝟐𝟐
−𝟗𝟗.𝟏𝟏
x ≈ {10.986, -2.4149}Extraneous Solution; Negative solution does not make sense!
x ≈ 11.0 secondsNearest tenth
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Problem #31 (Quadratic Equation/Problem Solving)
Hitting the ground:Height = 0
-16t2 + 336t + 112 = 0-16 -16 -16 -16
1t2 – 21t – 7 = 0
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Problem #31 CONT…Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
𝒙𝒙 = −(−𝟐𝟐𝟏𝟏)± (−𝟐𝟐𝟏𝟏)𝟐𝟐−𝟒𝟒(𝟏𝟏)(−𝟕𝟕)𝟐𝟐(𝟏𝟏)
𝒙𝒙 =𝟐𝟐𝟏𝟏 − 𝟒𝟒𝟔𝟔𝟗𝟗
𝟐𝟐,𝟐𝟐𝟏𝟏 + 𝟒𝟒𝟔𝟔𝟗𝟗
𝟐𝟐
Solve by quadratic formula!1t2 – 21t – 7 = 0
So a = 1, b = (-21), c = (-7)
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Problem #31 CONT…
x ≈ {-0.3282, 21.3282}Extraneous Solution; Negative solution does not make sense!
x ≈ 21.3 secondsNearest tenth
𝒙𝒙 =𝟐𝟐𝟏𝟏 − 𝟒𝟒𝟔𝟔𝟗𝟗
𝟐𝟐,𝟐𝟐𝟏𝟏 + 𝟒𝟒𝟔𝟔𝟗𝟗
𝟐𝟐
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Problem #32 (Vertex Problem)
Find vertex!Maximum value of parabola = Vertex (h, k).Vertex formula:t = − 𝒃𝒃
2𝒂𝒂(x-coordinate)
h(t) = y-coordinate
a = -16, b = 64
t = − 𝟔𝟔𝟒𝟒2 −𝟏𝟏𝟔𝟔
= 6432
= 2The arrow reaches the maximum height after 2 seconds.
h(t) = -16t2 + 64th(2) = -16(2)2 + 64(2)= 64 feet (Maximum height)
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Problem #33 (Graphing Quadratics)
Vertex formula:x = − 𝒃𝒃
2𝒂𝒂(x-coordinate)
f(x) = y-coordinate
x = − 𝟐𝟐2 𝟏𝟏
= −22
= −𝟏𝟏
f(x) = 1x2 + 2x – 3 ;a = 1, b = 2
Vertex:(-1, -4)
Since a = 1 (positive)Opens UPWARD!
f(-1) = (-1)2 + 2(-1) – 3 = -4
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Problem #33 (Sketch Graph):
Vertex:(-1, -4)
(-1, -4)
Touches x-axis 2 times;2 x-intercepts!!
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Problem #33 CONT…
y-intercept: x = 0x-intercept: y = 0
y-intercept: x = 0f(0) = 02 + 2(0) – 3 = – 3
y-intercept:(0, –3)
x-intercept: y = 0x2 + 2x – 3 = 0
x2 + 2x – 3 = 0(x + 3)(x – 1) = 0
Solve by factoring!
x = {-3, 1}x-intercepts: (-3, 0), (1, 0)
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Problem #33 CONT…
Vertex: (-1, -4)y-intercept: (0, -3)x-intercepts: (-3, 0), (1, 0)Opens UPWARD!
(-1, -4)
(-3, 0) (1, 0)
(0, -3)
Line of Symmetry:x = -1 (Vertical)x = h of vertex (h, k)
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(-1, -4)
(-3, 0)(1, 0)
(0, -3)
Problem #33
Solution:
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Domain and Range for all parabolas
DOMAIN: (-∞, ∞) OR All real numbers
k
∞
-∞
k
RANGE: [k, ∞) RANGE: (-∞, k]
UPDOWN
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Find Domain and Range for #33
x = - ∞Goes to left forever!
x = ∞Goes to right forever!
y = ∞Goes up forever
RANGE: [-4, ∞)
(-1, -4)
DOMAIN: (-∞, ∞) OR All real numbers
y = -4
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Range Visualized:
y = -4 y = ∞Goes up forever
RANGE: [-4, ∞)
(-1, -4)
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Problem #34 (Graphing Quadratics)
Vertex formula:x = − 𝒃𝒃
2𝒂𝒂(x-coordinate)
f(x) = y-coordinate
x = − −𝟒𝟒2 𝟐𝟐
= + 44
= 𝟏𝟏
F(x) = 2x2 – 4x + 5;a = 2, b = – 4
Vertex:(1, 3)
Since a = 2 (positive)Opens UPWARD!
F(1) = 2(1)2 – 4(1) + 5 = 3
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Problem #34 (Sketch Graph):
Vertex:(1, 3)
(1, 3)
Does not touch x-axis;No x-intercepts!!
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Problem #34 CONT…
y-intercept: x = 0x-intercept: y = 0
y-intercept: x = 0F(0) = 2(0)2 – 4(0) + 5 = 5
y-intercept:(0, 5)
x-intercept: y = 02x2 – 4x + 5 = 0
2x2 – 4x + 5 = 0
Solve by quadratic formula!2x2 – 4x + 5 = 0
So a = 2, b = (-4), c = 5
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Problem #34 CONT…Use Quadratic Formula:
𝒙𝒙 = −𝒃𝒃± 𝒃𝒃𝟐𝟐−𝟒𝟒𝒂𝒂𝒄𝒄𝟐𝟐𝒂𝒂
; Just plug in a, b, c
Quadratic Equation to solve:2x2 – 4x + 5 = 0
So a = 2, b = (-4), c = 5
𝒙𝒙 = −(−𝟒𝟒)± (−𝟒𝟒)𝟐𝟐−𝟒𝟒(𝟐𝟐)(𝟓𝟓)𝟐𝟐(𝟐𝟐)
𝒙𝒙 =𝟒𝟒 ± −𝟐𝟐𝟒𝟒
𝟒𝟒
NO SOLUTION! NO x-intercepts!−𝟐𝟐𝟒𝟒 = UNDEFINED!
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Problem #34 CONT…
Vertex: (1, 3)y-intercept: (0, 5)Another point: (2, 5)x-intercepts: NONE!Opens UPWARD!
(1, 3)
(0, 5)
Line of Symmetry:x = 1 (Vertical)
x = h of vertex (h, k)
(2, 5)
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Problem #34 Solution:
(1, 3)
(0, 5)
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Find Domain and Range for #34
x = - ∞Goes to left forever!
x = ∞Goes to right forever!
DOMAIN: (-∞, ∞) OR All real numbers
y = ∞Goes up forever
RANGE: [3, ∞)
(1, 3)y = 3
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Range Visualized:
y = 3 y = ∞Goes up forever
RANGE: [3, ∞) (1, 3)
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Problem #35 (Graphing Quadratics)
Vertex form:f(x) = a(x – h)2 + kVertex = (h, k)x-coordinate: Opposite Signy-coordinate: Copy!f(x) = 1(x + 2)2 – 5
Vertex:(-2, -5)
Since a = 1 (positive)Opens UPWARD!
Vertex = (–2, –5)
Line of Symmetry:x = -2 (Vertical)
x = h of vertex (h, k)
![Page 114: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/114.jpg)
Problem #35 CONT…
Vertex:(-2, -5)Opens UPWARD!
(-2, -5)Line of Symmetry:
x = -2 (Vertical)x = h of vertex (h, k)
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Problem #35 Solution:
(-2, -5)
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Find Domain and Range for #35
x = - ∞Goes to left forever!
x = ∞Goes to right forever!
DOMAIN: (-∞, ∞) OR All real numbers
y = ∞Goes up forever
RANGE: [-5, ∞)
(-2, -5)y = -5
![Page 117: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/117.jpg)
Range Visualized:
y = -5 y = ∞Goes up forever
RANGE: [-5, ∞)
(-2, -5)
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Problem #36 (Graphing Quadratics)
Vertex form:f(x) = a(x – h)2 + kVertex = (h, k)x-coordinate: Opposite Signy-coordinate: Copy!f(x) = -1(x – 3)2 + 0
Vertex:(-2, -5)
Since a = -1 (negative)Opens DOWNWARDS!
Vertex = (3, 0)
Line of Symmetry:x = 3 (Vertical)x = h of vertex (h, k)
![Page 119: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/119.jpg)
Problem #36 CONT…
Vertex:(3, 0)Opens DOWN!
(3, 0)
Line of Symmetry:x = 3 (Vertical)x = h of vertex (h, k)
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Problem #36 Solution
(3, 0)
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Find Domain and Range for #36
x = - ∞Goes to left forever!
x = ∞Goes to right forever!
DOMAIN: (-∞, ∞) OR All real numbers
RANGE: (−∞, 0]
(3, 0)
y = 0
y = −∞Goes down forever
![Page 122: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/122.jpg)
Range Visualized:
(3, 0)
RANGE:(−∞, 0]
y = 0y = −∞Goes down forever
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Problem #37 (Linear Graphing)
x y(0,
0)
y-intercept (x = 0):x + 2y = 80 + 2y = 8
2y = 8
y = 4 4)
x-intercept (y = 0):x + 2y = 8
x + 2(0) = 8
x = 8
(8,
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Problem #37 CONT…
(0, 4)
(8, 0)
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Problem #37 Solution:
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Problem #38 (Linear Graphing)
x y(0,(4,
x = 0:y = 𝟑𝟑
𝟒𝟒� 𝟏𝟏 + 3
Make x-y table!
y = 33)
x = 4:y = 𝟑𝟑
𝟒𝟒� 𝟒𝟒 + 3
y = 66)
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Problem #38 CONT…
(0, 3)
(4, 6)
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Solution for #38
(0, 3)
(4, 6)
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Problem #38 (Alternative Method)
y = 𝟑𝟑𝟒𝟒𝒙𝒙 + 𝟑𝟑
y-intercept:(0, 3)
y = mx + b (Slope-intercept form)y-intercept = (0, b)Slope = m
Slope = 𝟑𝟑𝟒𝟒
Up 3, Right 4
![Page 130: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/130.jpg)
Problem #38 Alternative CONT…
(0, 3)
(4, 6)
UP 3
y-intercept:(0, 3)
Slope = 𝟑𝟑𝟒𝟒
Up 3, Right 4
RIGHT 4
![Page 131: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/131.jpg)
Solution for #38
(0, 3)
(4, 6)
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Find Domain and Range for #38
x = - ∞Goes to left forever!
x = ∞Goes to right forever!
DOMAIN/RANGE: (-∞, ∞) OR All real numbers
y = ∞Goes up forever
y = −∞Goes down forever
![Page 133: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/133.jpg)
Problem #39 (Linear Graphing)Find intercepts!
x y(0,
0)
y-intercept (x = 0):-5x + 3y = -15-5�0 + 3y = -15
3y = -15
y = -5 -5)
x-intercept (y = 0):-5x + 3y = -15
-5x + 3(0) = -15
x = 3
(3,
-5x = -15
![Page 134: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/134.jpg)
Problem #39 CONT…
(0, -5)
(3, 0)
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(0, -5)
(3, 0)
Solution for #39
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Problem #40 (Linear Graphing)
Horizontal line; y = k
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Solution for #40
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Problem #41 (Linear Graphing)Vertical line; x = k
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Solution for #41
![Page 140: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/140.jpg)
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 –7y = -3x – 28
![Page 141: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/141.jpg)
Problem #43 (Function Notation)
Slope = 𝒚𝒚𝟐𝟐 − 𝒚𝒚𝟏𝟏𝒙𝒙𝟐𝟐 − 𝒙𝒙𝟏𝟏
=𝟒𝟒𝟑𝟑 −𝟏𝟏𝟏𝟏
𝟗𝟗 −𝟏𝟏
(9, 43) and (1, 11)(𝒙𝒙𝟐𝟐, 𝒚𝒚𝟐𝟐) and (𝒙𝒙𝟏𝟏, 𝒚𝒚𝟏𝟏)
Point-slope form: y – y1 = m(x – x1)
= 𝟑𝟑𝟐𝟐𝟏𝟏
= 𝟒𝟒
Choose (1, 11) for (𝒙𝒙𝟏𝟏, 𝒚𝒚𝟏𝟏) and m = 4.
![Page 142: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/142.jpg)
Choose (1, 11) for (𝒙𝒙𝟏𝟏, 𝒚𝒚𝟏𝟏) and m = 4.
Problem #43 CONT…
y – 11 = 4(x – 1)y – 11 = 4x – 4
+11 +11y = 4x + 7
f(x) = 4x + 7
![Page 143: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/143.jpg)
Problem #44 (Graphing Inequalities)
Notes:<, >: Dotted Line<, >: Solid Line
Notes:y > mx + bShade Above Liney < mx + bShade Below Line
Graph:y < 2x + 6; Dotted Line/Belowy > x – 8; Solid Line/Above
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Problem #44CONT…
Graph:y < 2x + 6; Dotted Line/Belowy > x – 8; Solid Line/Above
For y < 2x + 6:y-intercept = (0, 6)Slope = 2(Go up 2, over by 1)
(0, 6)
(1, 8)
For y > 1x – 8:y-intercept = (0, -8)Slope = 1(Go up 1, over by 1)
(0, -8)(1, -7)
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Problem #44 FINAL SOLUTION
(0, 6)
(1, 8)
(0, -8)(1, -7)
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Problem #44 Solution
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Problem #45 (Systems of Equations)
Let x = number of adult ticketsLet y = number of senior citizens tickets
Set-Up:x + y = 491 tickets$25x + $13y = $8195
Solve for y (Substitution):x + y = 491
y = (491 – x)
Solve: 25x + 13(491 – x) = 8195
25x + 6383 – 13x = 8195
−𝒙𝒙−𝒙𝒙
![Page 148: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/148.jpg)
Problem #45 CONT…
25x + 6383 – 13x = 8195
12x + 6383 = 8195−𝟔𝟔𝟑𝟑𝟏𝟏𝟑𝟑 −𝟔𝟔𝟑𝟑𝟏𝟏𝟑𝟑
12x = 181212 12
x = 151 adult tickets
y = 491 – xy = 491 – 151= 340 senior citizen tickets
![Page 149: MAT 1033C Final Exam Workshop - Valencia College...Table of Contents • Useful Formulas • Rational Expressions/Equations – #1, 2, 3, 4, 5, 6, 7, 8, 9, 47, 48, 49 • Radicals](https://reader036.vdocument.in/reader036/viewer/2022070809/5f0828587e708231d4209fd4/html5/thumbnails/149.jpg)
Problem #46 (Systems of Equations)
Let x = pounds of trail mixLet y = pounds of cashew
Set-Up:x + y = 75 pounds$5x + $15y = ($13)(75)
Solve (elimination):x + y = 75
5x + 15y = 975
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Problem #46 CONT….
x + y = 755x + 15y = 975
( )-5
We will eliminate x
-5x – 5y = -3755x + 15y = 975
10y = 60010 10
y = 60 pounds of cashew
x + y = 75x + 60 = 75
−𝟔𝟔𝟏𝟏 −𝟔𝟔𝟏𝟏
x = 15 pounds of trail mix
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Problem #47 (Rational Equations)
1(𝒙𝒙 + 𝟒𝟒)
−7
𝒙𝒙 − 𝟒𝟒=
4(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟒𝟒)
Factor: x2 – 16 (x + 4)(x – 4)
Restrictions: Denominator ≠ 0
Solve:x + 4 ≠ 0
x ≠ -4
Solve:x – 4 ≠ 0
x ≠ 4
Drop all solutionswhere x = {-4, 4}
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Problem #47 CONT…
1(𝒙𝒙 + 𝟒𝟒)
−7
𝒙𝒙 − 𝟒𝟒=
4(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟒𝟒)
LCD = (x + 4)(x – 4)
� (𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟒𝟒) � (𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟒𝟒) � (𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟒𝟒)
1(x – 4) – 7(x + 4) = 4
1x – 4 – 7x – 28 = 4
– 6x – 32 = 4+𝟑𝟑𝟐𝟐 +𝟑𝟑𝟐𝟐
– 6x = 36-6 -6
x = – 6No extraneous solutions!
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Problem #48 (Rational Equations)
Restrictions: Denominator ≠ 0
x ≠ 0
Drop all solutionswhere x = {0}
1 + 1𝑥𝑥
= 12𝑥𝑥𝑥𝑥
Factor: x2 xx
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Problem #48 CONT…
1 + 1𝑥𝑥
= 12𝑥𝑥𝑥𝑥
� 𝒙𝒙𝒙𝒙 � 𝒙𝒙𝒙𝒙 � 𝒙𝒙𝒙𝒙
x2 + 1x = 12
LCD = x2 (xx)
Make equation = 0
x2 + 1x – 12 = 0 Factor trinomial!
(x + 4)(x – 3)= 0 x = {-4, 3}No extraneous solutions!
−𝟏𝟏𝟐𝟐 −𝟏𝟏𝟐𝟐
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Problem #49 (Rational Equations)
(𝑥𝑥 + 5)(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟏𝟏)
−5
𝒙𝒙 − 𝟏𝟏 (𝒙𝒙 − 𝟏𝟏)=
(𝑥𝑥 − 5)(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟏𝟏)
Factor: x2 + 3x – 4 (x + 4)(x – 1)x2 – 2x + 1 (x – 1)(x – 1)
Restrictions: Denominator ≠ 0Solve:x + 4 ≠ 0
x ≠ -4
Solve:x – 1 ≠ 0
x ≠ 1
Drop all solutionswhere x = {-4, 1}
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Problem #49 CONT…(𝑥𝑥 + 5)
(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟏𝟏)−
5𝒙𝒙 − 𝟏𝟏 (𝒙𝒙 − 𝟏𝟏)
=(𝑥𝑥 − 5)
(𝒙𝒙 + 𝟒𝟒)(𝒙𝒙 − 𝟏𝟏)
LCD = (x + 4)(x – 1)(x – 1)� 𝒙𝒙 + 𝟒𝟒𝒙𝒙 − 𝟏𝟏𝒙𝒙 − 𝟏𝟏
� 𝒙𝒙 + 𝟒𝟒𝒙𝒙 − 𝟏𝟏𝒙𝒙 − 𝟏𝟏
� 𝒙𝒙 + 𝟒𝟒𝒙𝒙 − 𝟏𝟏𝒙𝒙 − 𝟏𝟏
𝑥𝑥 + 5 𝒙𝒙 − 𝟏𝟏 − 5 𝒙𝒙 + 𝟒𝟒 = (𝑥𝑥 − 5)(𝒙𝒙 − 𝟏𝟏)
FOIL! DISTRIBUTE! FOIL!
x2 – 1x + 5x – 5 – 5x – 20 = x2 – 1x – 5x + 5
x2 – 1x + 5x – 5 – 5x – 20 = x2 – 1x – 5x + 5
x2 – 1x – 25 = x2 – 6x + 5
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Problem #49 CONT…
x2 – 1x – 25 = x2 – 6x + 5−𝒙𝒙𝟐𝟐 −𝒙𝒙𝟐𝟐
– 1x – 25 = – 6x + 5+𝟔𝟔𝒙𝒙 +𝟔𝟔𝒙𝒙
5x – 25 = 5+𝟐𝟐𝟓𝟓 +𝟐𝟐𝟓𝟓
5x = 305 5
x = 6No extraneous solutions!
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Problem #50 (Compound Inequalities)
−∞ ∞
x < 3 (Shade left)x > -2 (Shade right)
][−∞ ∞
[ ]
[-2, 3]
Intersection ONLY!
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Problem #51 (Compound Inequalities)
11 < 5𝟐𝟐x + 6 < 31 � 𝟐𝟐 � 𝟐𝟐 � 𝟐𝟐 � 𝟐𝟐
22 < 5x + 12 < 62−𝟏𝟏𝟐𝟐−𝟏𝟏𝟐𝟐 −𝟏𝟏𝟐𝟐
10 < 5x < 505 5 5
2 < x < 10x is between2 and 10!
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Problem #51 (Compound Inequalities)
2 < x < 10; x is between 2 and 10!
−∞ ∞[ )
[2, 10) Remember:<, >: []<, >: ()
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Problem #52 (Compound Inequalities)
x + 4 < 1−𝟒𝟒 −𝟒𝟒
x < -3
-4x < 4-4 -4
x > -1Sign Flipped!
x < -3 and x > -1Intersection
ONLY!
−∞ ∞) (
There is no intersection/overlap! NO SOLUTION!
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Problem #52 Solution
−∞ ∞
Blank Number Line∅
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Problem #53 (Compound Inequalities)
−∞ ∞] [
(−∞, 4] ∪ [6, ∞)(−∞, 4] ∪ [6, ∞)
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Problem #54 (Compound Inequalities)
−∞ ∞) )
Beginning (left) to End (right)
−∞ ∞
)(−∞, 7)
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Problem #55 (Compound Inequalities)
-5x + 1 > 11−1 −1
-5x > 10-5 -5
x < -2Sign Flipped!
3x + 3 > -9−3 −3
3x > -123 3
x > -4 x < -2 or x > -4Beginning to end!
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Problem #55 CONT…
x < -2 or x > -4Beginning to end!
−∞ ∞][
Beginning (left) to End (right)
−∞ ∞
( )(−∞, ∞)
All real numbers!
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Problem #55 CONT…
−∞ ∞
( )(−∞, ∞)
All real numbers!
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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
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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!
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Helpful Study Tips:6) Visit the Math Connections for additional support and resources!Study a little each day, DO NOT CRAM!!
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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!
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Now go study and do well on your final exam!