Nierówności z wartością bezwzględną

Nierówności z wartością bezwzględną

Cztery typy nierówności z wartością bezwzględną oraz ich rozwiązania:

• \left|x-1\right|\left.\right.\textcolor{#005FD4}{<}2 ∣x − 1∣<  2
  x-1\textcolor{#005FD4}{<}2 x − 1  <  2 i x-1\textcolor{#005FD4}{>}-2 x − 1  >  −2
  x\textcolor{#005FD4}{<}3 x  <  3 i x\textcolor{#005FD4}{>}-1 x  >  −1
  x\in \left(-1;3\right) x  ∈  (−1; 3)

• \left|x-1\right|\left.\right.{\left.\right.\textcolor{#005FD4}{{\fontfamily{ams}\leqslant }}}\left.\right.2 ∣x − 1∣⩽ 2
  x-1\left.\right.\textcolor{#005FD4}{{\fontfamily{ams}\leqslant }}\left.\right.2 x − 1 ⩽ 2 i x-1\left.\right.\textcolor{#005FD4}{{\fontfamily{ams}\geqslant }}\left.\right.-2 x − 1 ⩾ − 2
  x\left.\right.\textcolor{#005FD4}{{\fontfamily{ams}\leqslant }}\left.\right.3 x ⩽ 3  i x\left.\right.\textcolor{#005FD4}{{\fontfamily{ams}\geqslant }}\left.\right.-1 x ⩾ − 1
  x\in \left\langle {-1;3}\right\rangle x  ∈  ⟨−1; 3⟩

• \left|x-1\right|\left.\right.\left.\right.\textcolor{#75A600}{>}2 ∣x − 1∣>  2
  x-1\textcolor{#75A600}{>}2 x − 1  >  2 lub x-1\textcolor{#75A600}{<}-2 x − 1  <  −2
  x\textcolor{#75A600}{>}3 x  >  3 lub x\textcolor{#75A600}{<}-1 x  <  −1
  x\in \left(-\infty ;-1\right)\left.\right.\left.\right.\textcolor{#75A600}{\cup }\left.\right.\left(3;\infty \right) x  ∈  (−∞; −1)∪ (3; ∞)

• \left|x-1\right|\left.\right.{\left.\right.\textcolor{#75A600}{{\fontfamily{ams}\geqslant }}}\left.\right.2 ∣x − 1∣⩾ 2
  x-1\left.\right.\textcolor{#75A600}{{\fontfamily{ams}\geqslant }}\left.\right.2 x − 1 ⩾ 2 lub x-1\left.\right.\textcolor{#75A600}{{\fontfamily{ams}\leqslant }}\left.\right.-2 x − 1 ⩽ − 2
  x\left.\right.\textcolor{#75A600}{{\fontfamily{ams}\geqslant }}\left.\right.3 x ⩾ 3 lub x\left.\right.\textcolor{#75A600}{{\fontfamily{ams}\leqslant }}\left.\right.-1 x ⩽ − 1
  x\in \left({-\infty ;-1}\right\rangle \left.\right.\left.\right.\left.\right.\textcolor{#75A600}{\cup }\left.\left\langle {3;\infty }\right)\right. x  ∈  (−∞; −1⟩∪ ⟨3; ∞)