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Solving Inequalities
PAES Preparation
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Solve x + 3 < 8x < 8 3
x < 5
m eans that whatever value xhas, it m ust be less than 5.
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Nu
mbers less than 5 are to the leftof 5 on the nu m ber line.
0 5 10 15-20 -15 -10 -5-25 20 25
If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.
There are also nu m bers in between the integers, like2.5, 1/2, -7.9, etc.
The num ber 5 would not be a correct answer,because 5 is not less than 5.
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2x + 7 32x 3 7
2x -4
x -2
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N um bers greater than -2 are to theright of -2 on the nu m ber line.
0 5 10 15-20 -15 -10 -5-25 20 25
If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.
There are also nu m bers in between the integers,like -1/2, 0.2, 3.1, 5.5, etc.
The num ber -2 w ould also be a correct answer,because of the phrase, or equal to.
-2
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W here is -1.5 on the nu m ber line?Is it greater or less than -2?
0 5 10 15-20 -15 -10 -5-25 20 25
-1.5 is between -1 and -2.-1 is to the right of -2.So -1.5 is also to the right of -2.
-2
-1.5 > -2
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Solve an Inequality3w + 5 < 13 - w
W e will use the sa m e steps that we did withequations, if a nu m ber is added to the variable, weadd the opposite sign to both sides:
3w + w < 13 - 5
4w < 8
w < 2
All nu m bers lessthan 2 are
solutions to thisproble m!
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M ore Exa m ples18 + r -2r
r + 2r -183r -18
r -6
All nu m bers fro m -6 and up (including
-6) m ake this proble m true !
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M ore Exa m ples-5 3x 2 16
-5 + 2 3x 16 + 2-3 3x 18
-1 x 6All nu m bers fro m -1 and up to 6 (including
-1 and 6) m ake this proble m true !
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R ules of Inequalitiesa > b
Is 2a > 2b? Alw ay s??Is - 2a > -2b Alw ay s?
If 4 > 3 is -8 > -6
All nu m bers greater than 0 m ake this
proble m true !
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M ore Exa m ples4 - 3 y 1
-3y 1 - 4-3y -3
y -3/-3 or y 1All nu m bers fro m 1 down (including 1)
m ake this proble m true ! NO
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Check y 14 - 3 y 1
4 3(0) 1NO
y -3/-3 or y 1All nu m bers fro m 1 up (including 1)
4 3(2) = -2 which is less than 1
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Quadratic Inequalitiesx2 + 3x - 10 0
(x + 5)(x 2) 0Below -5 Between -5 and 2 Above 2
+ - +
-5 x 2All nu m bers between -5 and 2 m ake this
proble m true
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Quadratic Inequalitiesx2 + 10x - 11 0(x + 11)(x 1) 0
Below -11 Between -11 and 1 Above 1
+ - +
x -11 or x 1All nu m bers less than -11 and greater than
1 m ake this proble m true
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Questions1. 5x + 3 < 182. 3x 2 > x + 53. 5(2 + x) > 31 2x4. 10 3x < 22
5. -3 < 2x 5 < 126. x 2 6x 16 > 0