Question 230602
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Next time use the 'pipe' symbol, Shift-Backslash, to indicate absolute value.  Square brackets are just like parentheses, so cause confusion when used for absolute value.


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 4\ -\ |3k\ +\ 1|\ <\ 2]


Add -4 to both sides:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -\ |3k\ +\ 1|\ <\ -2]


Multiply by -1 (remember to change the sense of the inequality)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |3k\ +\ 1|\ >\ 2]


Apply the definition of absolute value:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ |x|\ =\ \left\{\ \ x\text{ if }x\ \geq\ 0\cr-x\text{ if }x\ <\ \,0\right]


So:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 3k\ +\ 1\ >\ 2\ \Rightarrow\ k\ >\ \frac{1}{3}]


Or


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ -(3k\ +\ 1)\ >\ 2 \Rightarrow\ k\ <\ -1]


So the solution set is *[tex \LARGE \{k\,|\,k\,\in\,\R,k\ <\ -1\}\ \large{\cup}\LARGE\ \{k\,|\,k\,\in\,\R,k\ >\ \frac{1}{3}\}]


Or in interval notation:  *[tex \LARGE \left(-\infty,\,-1\right)\ \large\cup\LARGE\ \left(\frac{1}{3},\,\infty\right)]


John
*[tex \LARGE e^{i\pi} + 1 = 0]
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