Solving Inequalities Using Addition and Subtraction

Lessons 3-1 and 3-2

Addition Property of Inequalities – If any number is

________________ to each side of a true ___________________,

the resulting inequality is also ________________. Example A 3 -5

3 + 2 -5 + 2

_____ _____

added

equation

true

>

>

>5 -3

Example B n – 12 < 65

n – 12 +12 < 65 + 12

Work inequalities horizontally.n < 77

“Open” circle (unshaded) at 77, then shade to the left (because it is “less than”). This means any number smaller than 77 is a solution to the inequality

{ n│ n < 77}This is called “set-builder notation.” It would be read as “n such that n is less than 77.”

Example C k – 4 > 10

k – 4 + 4 > 10 + 4

k > 14

Adding the same number to each side of an inequality does not change the direction of the inequality.

{k │k > 14 }

Set builder notation is always placed inside of braces.

Graphing on the Number Line (A Quick Review)

Great than or equal to (≥) and less than or equal to (≤) uses a filled in (or closed) circle then shade the line in the same direction the symbol is pointing.

Great than (>) and less than (<) uses an unshaded (or open) circle then shade the line in the same direction the symbol is pointing.

12 + 9 y – 9 + 9

y

21y

21

≥

≤

≥

Add

y│y ≤ 21

Subtraction Property of Inequality – If any number is

___________

from ________ side of a true inequality, the resulting

inequality is also _______.

subtracted

each

true

subtractq + 23 - 23 14 - 23<

q < -9

{q│q < –9}

x – 2 < 8

x – 2 + 2 < 8 + 2

x < 10

The symmetric property does not work for inequalities, so if you “turn the inequality around” you have to change the sign, too.

{ x │ x < 10}

m + 15 – 15 ≤ 13 - 15

m ≤ –2

{ m │ m ≤ –2 }

Variables on Both Sides

Example H 12n – 4 ≤ 13n

12n –12n – 4 ≤ 13n –12n

– 4 ≤ n

n ≥ – 4

{ n │n ≥ – 4}

Example I 3p – 6 ≥ 4p

3p – 3p – 6 ≥ 4p – 3p

– 6 ≥ p

p ≤ – 6

{ p │ p ≤ – 6 }

Example J 5x + 4 > 4x + 10

5x + 4 – 4x > 4x – 4x + 10

x + 4 > 10

x + 4 – 4 > 10 – 4

x > 6

{x│x > 6 }

Ex. K Seven time a number is greater than 6 times that number minus two.7x > 6x – 2

7x – 6x > 6x – 6x – 2

x > – 2

Ex. L Three times a number is less than two times that number plus 5. 3x < 2x + 5

3x – 2x < 2x – 2x + 5

x < 5