Russ P. answered • 12/07/14
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Patient MIT Grad For Math and Science Tutoring
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Because you have an absolute value sign, you have to solve two cases: the positive case [+(2x + 7)] and the negative case [-(2x + 7)]. And note that the + or - sign goes in front of the entire entry inside the absolute value | | signs. That's why I enclosed the entire entry inside parentheses.
Before solving these two cases I'll simplify your left-hand-side expression by moving the -3 across the > sign where it becomes a +3. So solving and graphing:
|(2x + 7)| > 13 .
If you graph this (where x-axis is horizontal) and the function y = |(2x + 7)| is vertical, you get a giant letter "V". Its vertex occurs at the (x,y) point (-3.5, 0). For all x > -3.5 it is a straight line y = (2x +7) (the positive case), whose slope is +2 and y-intercept of +7. For all x < -3.5 it is another straight line y = -(2x + 7), (the negative case), whose slope is -2 and y-intercept is -7.
I f you now graph the right-hand side, a horizontal line at y = +13. Then the solution graphically is all those x whose "V" points or y's are above that line. These are all x < -10 and all x > +3. Now we'll do it algebraically:
The positive case:
+(2x + 7) > 13
2x > 13 - 7 = 6
x > +3.
The negative case:
-(2x + 7) > 13
-2x -7 > 13
-2x > +20 as we bring the -7 over to the other side of > where it becomes +7 (note, you've added +7 to both sides, thereby moving it)
2x < -20 , when you multiply both sides by a negative you flip the > sign to a < sign.
x < -10
Thus the combined solution is that x < -10 or X > +3. In-between it fails your equation.
Same answer as in the graphical solution as you would expect. That tells you that you did it right. You can (if you want) also check it algebraically by plugging in X = -12, 0, and +5 into your original equation, and see that x=0 fails while the other 2 satisfy the original inequality.
