paes inequalities

8/7/2019 PAES Inequalities

1/16

Solving Inequalities

PAES Preparation


2/16


3/16

Solve x + 3 < 8x < 8 3

x < 5

m eans that whatever value xhas, it m ust be less than 5.


4/16

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.


5/16

2x + 7 32x 3 7

2x -4

x -2


6/16

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


7/16

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


8/16

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!


9/16

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 !


10/16

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 !


11/16

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 !


12/16

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


13/16

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


14/16

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


15/16

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


16/16

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

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