Chapter 6
Solving Linear Inequalities
Work towards isolating the variable…
Check to see if the inequality is completely simplified…
Can you distribute or combine like terms?
YES
NO
Simplify
1. Distribute if necessary
2. Combine like terms if possible
NEXT STEP
3(2x + 7) + 4x > 31
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Begin to isolate…�*Don’t forget that whatever you do to one side, you must do the other!
Are there variables on both sides of the equation?
YES
NO
Get variables together
Move one of the variable terms to the other side of the equation. Your only options are to add or subtract.
2x – 8 > 4 – 4x
+4x +4x
6x – 8 > 4
NEXT STEP
Continue Isolating…
Is there anything adding or subtracting to/from the variable?
YES
NO
GET RID OF IT!
Use the inverse operation, either addition or subtraction, to cancel out the number that is being added or subtracted. (Make it 0)
Ex: 2x + 3 > 11
–3 – 3
Ex: –4 + 5x > 16
+4 +4
Subtracting 3 cancels out the 3.
(Makes it 0)
Adding 4 cancels out the –4 .
(Makes it 0)
NEXT STEP
Continue Isolating…
Is there anything multiplying or dividing the variable?
YES
NO
GET RID OF IT!
Use the inverse operation, either multiplication or division, to cancel out the number that is multiplying or dividing the variable. (Make it 1)
Ex:
Dividing by 2 cancels out 2. (makes it 1)
Ex:
3·
·3
Multiplying by 3 cancels out the 3.
(Makes it 1)
NEXT STEP
Your variable should be isolated
IT IS!
YAY!
Ex: x > 3
That’s my answer!
Umm…the variable is gone…
BTWs…this means that x can be any number that is greater than 3. There are infinite possibilities.
“No Solution” vs. “All Real #’s”
*Remember that your goal is to state what x can be. For example:
x > 3 says that x can be any number that is bigger than 3. For example: 3.1, 4, 5, or even 100,000.
3 > 10 doesn’t say that x can be 3 or 10, it just says “3 is greater than 10,” which is FALSE. Since it’s a false statement it means that x cannot be anything so we say…
NO SOLUTION!
4 < 8 doesn’t say compare x to 4 or 8. It just says “4 is less than or equal to 8,” which is TRUE! Since it’s a true statement it means that x can be anything so we say…
ALL REAL NUMBERS!
*NEW RULE ALERT*
Did you MULTIPLY or DIVIDE by a NEGATIVE to finish isolating x?
Yes
No
Then you must reverse the inequality sign!
x
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3
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NEXT STEP