Rozwiąż nierówność |2x-|1-3x|| < (bądź równe) 4.
Mathēmatikós
![\pmb1. \ 2x - | 1-3x| \leq 4 \\ \\ \cdot \ \ \ x \ \textless \ \frac{1}{3} \implies |1-3x| = 1-3x \\ \\ 2x - (1-3x) \leq 4 \\ \\ 2x - 1 +3x \leq 4 \\ \\ 5x \leq 5 \\ \\ x \leq 1 \\ \\ (-\infty, 1] \ \cap \ (-\infty ,\frac{1}{3} ) = \boxed{(- \infty, \frac{1}{3}) }](https://tex.z-dn.net/?f=%5Cpmb1.+%5C+2x+-+%7C+1-3x%7C+%5Cleq+4+%5C%5C+%5C%5C+%5Ccdot+%5C+%5C+%5C+x+%5C+%5Ctextless+%5C+%5Cfrac%7B1%7D%7B3%7D+%5Cimplies+%7C1-3x%7C+%3D+1-3x+%5C%5C+%5C%5C+2x+-+%281-3x%29+%5Cleq+4+%5C%5C+%5C%5C+2x+-+1+%2B3x+%5Cleq+4+%5C%5C+%5C%5C+5x+%5Cleq+5+%5C%5C+%5C%5C+x+%5Cleq+1+%5C%5C+%5C%5C+%28-%5Cinfty%2C+1%5D+%5C+%5Ccap+%5C+%28-%5Cinfty+%2C%5Cfrac%7B1%7D%7B3%7D+%29+%3D+%5Cboxed%7B%28-+%5Cinfty%2C+%5Cfrac%7B1%7D%7B3%7D%29+%7D++ "\pmb1. \ 2x - | 1-3x| \leq 4 \\ \\ \cdot \ \ \ x \ \textless \ \frac{1}{3} \implies |1-3x| = 1-3x \\ \\ 2x - (1-3x) \leq 4 \\ \\ 2x - 1 +3x \leq 4 \\ \\ 5x \leq 5 \\ \\ x \leq 1 \\ \\ (-\infty, 1] \ \cap \ (-\infty ,\frac{1}{3} ) = \boxed{(- \infty, \frac{1}{3}) }")
![\cdot \ \ \ x \geq \frac{1}{3} \implies |1-3x| = -(1-3x) = 3x-1 \\ \\ 2x-(3x-1) \geq -4 \\ \\ 2x - 3x + 1 \geq -4 \\ \\ -x \geq -5 \\ \\ x \leq 5 \\ \\ \lbrack \frac{1}{3}, \infty) \ \cap \ (-\infty, 5] = \boxed{[ \frac{1}{3}, 5]} \\ \\ \\ \\ \lbrack - \frac{3}{5}, \frac{1}{3}) \ \cup \ [ \frac{1}{3}, 5] = \boxed{\boxed{[- \frac{3}{5}, 5] }}](https://tex.z-dn.net/?f=%5Ccdot+%5C+%5C+%5C+x++%5Cgeq++%5Cfrac%7B1%7D%7B3%7D+%5Cimplies+%7C1-3x%7C+%3D+-%281-3x%29+%3D+3x-1+%5C%5C++%5C%5C+2x-%283x-1%29++%5Cgeq+-4+%5C%5C++%5C%5C+2x+-+3x+%2B+1++%5Cgeq++-4++++%5C%5C++%5C%5C+-x++%5Cgeq+-5+%5C%5C++%5C%5C+x+%5Cleq+5+%5C%5C++%5C%5C+%5Clbrack++%5Cfrac%7B1%7D%7B3%7D%2C+%5Cinfty%29+%5C+%5Ccap+%5C+%28-%5Cinfty%2C+5%5D+++%3D+%5Cboxed%7B%5B+%5Cfrac%7B1%7D%7B3%7D%2C+5%5D%7D+++%5C%5C++%5C%5C++%5C%5C++%5C%5C+%5Clbrack+-+%5Cfrac%7B3%7D%7B5%7D%2C++%5Cfrac%7B1%7D%7B3%7D%29+%5C+%5Ccup+%5C+%5B+%5Cfrac%7B1%7D%7B3%7D%2C+5%5D+%3D+%5Cboxed%7B%5Cboxed%7B%5B-+%5Cfrac%7B3%7D%7B5%7D%2C+5%5D+%7D%7D+++ "\cdot \ \ \ x \geq \frac{1}{3} \implies |1-3x| = -(1-3x) = 3x-1 \\ \\ 2x-(3x-1) \geq -4 \\ \\ 2x - 3x + 1 \geq -4 \\ \\ -x \geq -5 \\ \\ x \leq 5 \\ \\ \lbrack \frac{1}{3}, \infty) \ \cap \ (-\infty, 5] = \boxed{[ \frac{1}{3}, 5]} \\ \\ \\ \\ \lbrack - \frac{3}{5}, \frac{1}{3}) \ \cup \ [ \frac{1}{3}, 5] = \boxed{\boxed{[- \frac{3}{5}, 5] }}")
![(-\infty, \infty) \ \cap \ [- \frac{3}{5}, 5] = \boxed{\boxed{\boxed{[- \frac{3}{5}, 5]}}}](https://tex.z-dn.net/?f=%28-%5Cinfty%2C+%5Cinfty%29+%5C+%5Ccap+%5C+%5B-+%5Cfrac%7B3%7D%7B5%7D%2C+5%5D+%3D+%5Cboxed%7B%5Cboxed%7B%5Cboxed%7B%5B-+%5Cfrac%7B3%7D%7B5%7D%2C+5%5D%7D%7D%7D+ "(-\infty, \infty) \ \cap \ [- \frac{3}{5}, 5] = \boxed{\boxed{\boxed{[- \frac{3}{5}, 5]}}}")
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