1<= |2x-1/x+5|<= 7

Lösung

1 \leq 2 x - \frac{1}{x} + 5 \leq 7

- \frac{6 + 38}{2} \leq x \leq - \frac{3 + 11}{2} or - \frac{2 + 6}{2} \leq x \leq \frac{1 - 3}{2} or \frac{38 - 6}{2} \leq x \leq \frac{11 - 3}{2} or \frac{6 - 2}{2} \leq x \leq \frac{1 + 3}{2}

Intervall-Notation

[- \frac{6 + 38}{2}, - \frac{3 + 11}{2}] \cup [- \frac{2 + 6}{2}, \frac{1 - 3}{2}] \cup [\frac{38 - 6}{2}, \frac{11 - 3}{2}] \cup [\frac{6 - 2}{2}, \frac{1 + 3}{2}]

Dezimale

- 6.08220 \dots \leq x \leq - 3.15831 \dots or - 2.22474 \dots \leq x \leq - 0.36602 \dots or 0.08220 \dots \leq x \leq 0.15831 \dots or 0.22474 \dots \leq x \leq 1.36602 \dots

Schritte zur Lösung

1 \leq 2 x - \frac{1}{x} + 5 \leq 7

Wenn

a \leq u \leq b

dann

a \leq u and u \leq b

1 \leq 2 x - \frac{1}{x} + 5 and 2 x - \frac{1}{x} + 5 \leq 7

1 \leq 2 x - \frac{1}{x} + 5 : x \leq - \frac{3 + 11}{2} or - \frac{2 + 6}{2} \leq x < 0 or 0 < x \leq \frac{11 - 3}{2} or x \geq \frac{6 - 2}{2}

2 x - \frac{1}{x} + 5 \leq 7 : - \frac{6 + 38}{2} \leq x \leq \frac{1 - 3}{2} or \frac{38 - 6}{2} \leq x \leq \frac{1 + 3}{2}

Kombiniere die Bereiche

(x \leq - \frac{3 + 11}{2} or - \frac{2 + 6}{2} \leq x < 0 or 0 < x \leq \frac{11 - 3}{2} or x \geq \frac{6 - 2}{2}) and (- \frac{6 + 38}{2} \leq x \leq \frac{1 - 3}{2} or \frac{38 - 6}{2} \leq x \leq \frac{1 + 3}{2})

Füge die sich überschneidenden Intervalle zusammen

- \frac{6 + 38}{2} \leq x \leq - \frac{3 + 11}{2} or - \frac{2 + 6}{2} \leq x \leq \frac{1 - 3}{2} or \frac{38 - 6}{2} \leq x \leq \frac{11 - 3}{2} or \frac{6 - 2}{2} \leq x \leq \frac{1 + 3}{2}

0 \leq 2 x - \frac{3}{4} - 3 x \leq 1

- 7 < 3 x + 2 \leq 5

- 3 < 2 + 4 x < 12

\frac{( 2 ^{1 - x} - 2 ^{x} + 1 )}{( 2 ^{x} - 1 )} \leq 0

2^{x} + 2^{- x + 1} - 3 < 0