SOLUTION: 4|2x+3|-7<9

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Question 1203651: 4|2x+3|-7<9
Found 3 solutions by ikleyn, MathLover1, greenestamps:
Answer by ikleyn(52788) About Me  (Show Source):
You can put this solution on YOUR website!
.
4|2x+3|-7 < 9
~~~~~~~~~~~~~~~~~~~~~~ 

They want you solve this inequality.


Move -7 to the right with changing the sign.  You will get

    4*|2x+3| < 9 + 7,  or

    4*|2x+3| < 16.


Divide both sides by 4.  You will get

    |2x+3| <4.


It means that

    -4 < 2x+3 < 4.


Move 3 from the central part to the left and to the right, changing the sign.  You will get  

    -4 - 3 < 2x < 4 - 3,

or

    -7 < 2x < 1.


Divide everything by 2.  You will get

    -3.5 < x < 0.5.


It is your ANSWER:  the solution set is  -3.5 < x < 0.5,  or the interval (-3.5,0.5),
which does not include the endpoints.

Solved, with complete explanations.

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To see many other similar and different SOLVED problems on absolute value inequalities,  look into the lesson
    - Solving absolute value inequalities
in this site.



Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

+4abs%282x%2B3%29-7%3C9
+4abs%282x%2B3%29%3C9%2B7
+abs%282x%2B3%29%3C16%2F4
+abs%282x%2B3%29%3C4
+2x%2B3%3C4
+2x%3C4-3
+2x%3C1
+x%3C1%2F2

+-%282x%2B3%29%3C4
+-2x-3%3C4
+-4-3%3C2x
+2x+%3E+-7
+x+%3E+-7%2F2

solution :
+-7%2F2%3Cx%3C1%2F2

Interval notation:
(+-7%2F2, +1%2F2)


Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The solutions shown by the other tutors are good formal mathematical solutions.

A different method for solving absolute value problems like this is to interpret

abs%28x-a%29%3Cb

as

"x is less than b units from a"

Applying that method to this problem....

4abs%282x%2B3%29-7%3C9
4abs%282x%2B3%29%3C16
abs%282x%2B3%29%3C4
abs%28x%2B1.5%29%3C2
abs%28x-%28-1.5%29%29%3C2

This says that x is less than 2 units from -1.5.

2 units to the left of -1.5 is -3.5; 2 units to the right of -1.5 is 0.5. So the solution is

-3.5 < x < 0.5