SOLUTION: solve each equations & inequality with absolute value a)|2x+1|-3=12 b)|3x+1|+2<6 c)|1/3x+7|+5>6

Question 1202532: solve each equations & inequality with absolute value
a)|2x+1|-3=12
b)|3x+1|+2<6
c)|1/3x+7|+5>6
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

I'll do part (a) to get you started.

We'll need the rule that |x| = k leads to x = k or x = -k, where k > 0.
Example: |x| = 7 leads to x = 7 or x = -7.
Both values are the same distance to zero on the number line.

|2x+1|-3 = 12
|2x+1| = 12+3
|2x+1| = 15
2x+1 = 15 or 2x+1 = -15 ... use the rule mentioned above
2x = 15-1 or 2x = -15-1
2x = 14 or 2x = -16
x = 14/2 or x = -16/2
x = 7 or x = -8

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Slightly another way to solve
Let w = |2x+1|

The equation
|2x+1|-3 = 12
becomes
w-3 = 12
which solves to
w = 15

That's another way to arrive at the step |2x+1| = 15
The remaining steps are the same as the previous section.

-----------------------------

Answer to part (a): x = 7 or x = -8

Hint for part (b): If |x| < k then -k < x < k, where k is positive.
Example: |x| < 5 means -5 < x < 5.

Hint for part (c): If |x| > k then x < -k or x > k, where k is positive.
Example: |x| > 12 means either x < -12 or x > 12.

Another hint: Draw out number lines to see why those formulas work the way they do.

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