Introduction: How to Solve an Absolute Value Inequality
Some of the first steps to solving a absolute value inequality is identifying if it will be AND or OR. This is told by the was the inequality sign is pointing. The sign here is a less than so it will be an AND. This is a 4 step problem.
Step 1: Getting the Absolute Value by Its Self
Now to start, you need to get the absolute value on one side alone. To do this correctly use the inverse operation on the addition portion. You need to add or subtract first always! Do this to its self; which cancels out and the other side of the inequality. Your new answer would be 24.
Step 2: Getting Absolute Value by Its Self Cont...
Now this is where some people struggle. Be careful not to distribute the 2 but instead, again, use the inverse operation and this time it will be dividing because the 2 is multiply. Again what you do to one side you must do to the other. Now as you can see the answer is 12. At this point if you don't have any more numbers on the outside of the absolute value bars like me, you have successfully gotten it alone!
Step 3: Getting 2 Answers
Being able to solve this correctly you will need 2 answers. To start this use the equation you have almost completed. The only time you can do this step is when the absolute value is isolated. Copy what it is but instead you need to make the whole number on the right a negative. As I mentioned before this will give you a range so you can plat it on a number line if needed.
Step 4: Getting Your Final Answer
Your last and final steps to solving this is to get x by its self on one side when the whole number should be on the other. For this problem I used the inverse operation by subtracting 14 to its self and the other side. And also for the second equation. Now you should get he final inequality!
Step 5: Checking Your Solutions
Now to see what kind of solution it is plug one of the answers in for the variable. This step is not always necessary. Inside the absolute value bars solve for what it says. For me it is -2 + 14.
Step 6: Checking Cont...
Next, multiply if needed.
Step 7: Checking Cont...
Step 8: Checking Cont...
And this works! 30 is not greater than 30 but since there is an equal to sign under the inequality this is true.
Step 9: Checking Your Solutions #2
And the same steps apply. Add.
Step 10: Checking Cont...
Multiply. But be careful, the absolute vaule of -12 is 12. That means it is 2 x 12.
Step 11: Checking Cont...
Step 12: Checking Cont...
And I got the same thing! It works. Sometimes you will end up with one working and the other not, this is called an Extraneous Solution.
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What if instead of 2 the factor was -2?
This would still work. You would divide over a negative 2 to get -12, and still solve for both 12 and -12.
Then you would do the same thing but it should be dived to itself as a -2.