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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