Exercise

Bestimme jeweils die Lösungsmenge L_x der folgen Bruchgleichungen. nprvmulticols abclist abc dfracx dfracx- abc dfracx+x+ dfracx+ abc dfracx+x- - dfracx-x- dfracx-x- abc dfracx-x+ - dfracx-x+ dfracx-x+ abc dfracx+ - dfracx- dfracx+-x^ abc dfracxx- - dfracxx- dfracx- abc dfracx- + dfracx dfracx+ abc dfracx-x - dfracx-x + dfracx+ abc dfracx - dfracx+ dfracx+ - dfracx+ abc dfrac.x-x+. - dfrac.x+. abc dfracx^+x +x+ x- abc dfracx^x^- - dfracx-x+ dfrac-x-x^ abc dfracx- - dfracx+ dfracx+x^- abc dfracx-x+ - dfracx-x+ dfracx+x^+x+ abc dfracx^-x+x^-x- dfracx^+x-x^-x- abc dfracx^-x+x^+x- dfracx^+x+x^+x- abc dfracx-x+ dfracxx+ abc dfracx + dfracx+x+ x+ abc dfracx^x- - dfracx-x abc . - dfracx-x+  dfracx+x abc dfracx+ + dfracx+ dfracx+ abc dfracxx- - dfracx+ dfracx- abc dfrac-x^x^- + dfrac+x^x^-x+  x abc dfracx+x^-x + dfracx-x^+x dfracx+x^- abclist nprvmulticols
newcommanddber[]ensuremathD_x R setminus qty# newcommandxloes[]x # && L_x left # right newcommandloes[][]&&L_# qty# abclist abc dber, al dfracx dfracx- &&|cdot xx- x- x &&|-x - x &&|: xloes-frac abc dber- al dfracx+x+ dfracx+ &&|cdot x+ x + &&|- x - &&|: xloes-frac abc dber al dfracx+x- - dfracx-x- dfracx-x- &&|cdot x- x+ - x- x - &&|text TU -x + x- &&|+x, + x &&|: xloes abc dber- al dfracx-x+ - dfracx-x+ dfracx-x+ &&|cdot x+ x- - x- x- &&|text TU - x- && |+ - x &&|: xloes-frac abc dber-, al dfracx+ - dfracx- dfracx+-x^ &&|cdot x+x- x- - x+ -x+ && |text TU - -x- && |+x, + xloes abc dber, al dfracxx- - dfracxx- dfracx- &&|cdot x-x- xx- - xx- x- &&|text TU -x x - &&|+x, + x &&|: xloesfrac abc dber-,, al dfracx- + dfracx dfracx+ &&|cdot xx-x+ xx+ + x-x+ xx- &&|text TU x^+x + x^- x^ - x &&| -x^, +, +x x &&|: xloesfrac abc dber-frac, al dfracx-x - dfracx-x + dfracx+ &&|cdot xx+ x+x- - x+x- + cdot x &&|text TU x^-x+x- -x^ + x - x + + x &&|text TU  x - &&|+, : xloesfrac abc dber-, -, -, al dfracx - dfracx+ dfracx+ - dfracx+ &&|cdot xx+x+x+ x+x+x+ - xx+x+ xx+x+ - xx+x+ &&|text TU x^ + x^ +x +x^ + x + - x^ -x^ - x x^ + x^ +x - x^ -x^ -x && |text TU x^ + x + x^ + x &&|-x^, -x -x &&|: xloes-frac abc dber-. al dfrac.x-x+. - dfrac.x+. &&|cdotx+. .x- - . x+. &&| text TU x - . x+. &&|-x, -. xloes-.-frac abc dber- al dfracx^+x +x+ x- &&|cdotx+ x^+x + x-x+ &&|text TU x^ +x + x^ + x - &&|-x^, -x, - x - &&|: x - && L_x , abc dber-, al dfracx^x^- - dfracx-x+ dfrac-x-x^ &&|cdotx^- x^ - x-x- --x &&|text TU x^ - x^+x - -+x && |text TU x- -+x &&| -x, + &&L_x textdber-, abc dber-, al dfracx- - dfracx+ dfracx+x^- &&|cdot x-x+ x+ - x- x + &&|text TU x + x + &&|-x,- &&L_x textdber-, abc dber-,- al dfracx-x+ - dfracx-x+ dfracx+x^+x+ &&|cdot x+x+ x+x- - x+x- x + &&|text TU x^ + x - - x^ -x + x + &&|text TU x + x + &&|-x,- -x uf:- x - &&L_x , abc Wir bestimmen den Definitionsbereich durch das Lösen der Gleichung x^-x- . Die Mitternachtsformel gibt uns die Lösungen al x_, frac pm sqrt+ frac pm x_ x_ -frac. Damit ist der Definitionsbereich dber-dfrac, . aldfracx^-x+x^-x- dfracx^+x-x^-x- uf cdotx^-x- x^-x+ x^ + x - uf -x^,-x,+ x^ -x + lf x-x- loes[x] abc Wie bei der vorherigen Teilaufgabe bestimmen wir den Definitionsbereich mit der Mitternachtsformel: al x_,  frac- pm sqrt+ frac- pm -pm &&dber-, Wir lösen die Bruchgleichung wie üblich: al dfracx^-x+x^+x- dfracx^+x+x^+x- ufcdot x^+x- x^ -x + x^ + x + uf -x^,+x,- x^ + x - lf x-x+ loes[x], abc dber-dfrac, - dfrac aldfracx-x+ dfracxx+ uf cdotx+x+ x+x- xx+ x^ +x - x^ + x uf -x^,-x x^ + x - uf : x^ + x - mf x_,  frac- pm sqrt + frac-pm   frac- pm loes[x],-frac abc dber-dfrac aldfracx + dfracx+x+ x+ uf cdot x+ xx+ + x+ x+x+ tu x^ + x + x + x^ +x + uf -x^,-x- x^ + x - mf x_, frac- pm sqrt+ frac- pm loes[x]-frac, frac abc dber aldfracx^x- - dfracx-x uf cdot x- x^+x x- uf -x,+ x^ + x + lf x+x+ loes-,- abc dber,-dfrac al. - dfracx-x+  dfracx+x uf cdot xx+ xx+ - xx- x+x+ tu x^ + x - x^ + x x^ + x + tu -x^ + x x^ + x+ uf +x^,-x x^ - x + mf x_, frac pm sqrt - frac pm loes,frac abc dber-,-, aldfracx+ + dfracx+ dfracx+ ufcdot x+x+x+ x+x+ + x+x+ x+x+ tu x^ +x + + x^ +x + x^ + x + tu x^ + x + x^+x + uf -x^,-x,- x^ -x - mf x_, frac pm sqrt+ frac pm loes-frac, abc dber-, aldfracxx- - dfracx+ dfracx- uf cdot x-x+ xx+ -x- x- tu x^ + x -x + x- tu x^ -x + x- uf-x,+ x^ -x + mf x_, frac pm sqrt- frac pm pm loes[x] abc dber-, aldfrac-x^x^- + dfrac+x^x^-x+  x tu dfrac-x^x+x- + dfrac+x^x-^ x uf cdot x+x-^ -x^x- + +x^x+ xx-^x+ tu x- -x^ +x^ +x + + x^ + x^ x^+xx^-x+ tu x^ + x x^ -x^ + x^ +x^-x^ + x uf -x^,-x x^ - x^ -x^ tu x^x-x+ loes-,, abcdber-,, aldfracx+x^-x + dfracx-x^+x dfracx+x^- tu dfracx+xx- + dfracx-xx+ dfracx+x+x- uf cdot xx-x+ x+^ + x-x- xx+ tu x^ + x + + x^ -x + x^ + x tu x^ -x + x^ + x uf -x^,-x x^ -x + lf x-x- loes abclist
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Bestimme jeweils die Lösungsmenge L_x der folgen Bruchgleichungen. nprvmulticols abclist abc dfracx dfracx- abc dfracx+x+ dfracx+ abc dfracx+x- - dfracx-x- dfracx-x- abc dfracx-x+ - dfracx-x+ dfracx-x+ abc dfracx+ - dfracx- dfracx+-x^ abc dfracxx- - dfracxx- dfracx- abc dfracx- + dfracx dfracx+ abc dfracx-x - dfracx-x + dfracx+ abc dfracx - dfracx+ dfracx+ - dfracx+ abc dfrac.x-x+. - dfrac.x+. abc dfracx^+x +x+ x- abc dfracx^x^- - dfracx-x+ dfrac-x-x^ abc dfracx- - dfracx+ dfracx+x^- abc dfracx-x+ - dfracx-x+ dfracx+x^+x+ abc dfracx^-x+x^-x- dfracx^+x-x^-x- abc dfracx^-x+x^+x- dfracx^+x+x^+x- abc dfracx-x+ dfracxx+ abc dfracx + dfracx+x+ x+ abc dfracx^x- - dfracx-x abc . - dfracx-x+  dfracx+x abc dfracx+ + dfracx+ dfracx+ abc dfracxx- - dfracx+ dfracx- abc dfrac-x^x^- + dfrac+x^x^-x+  x abc dfracx+x^-x + dfracx-x^+x dfracx+x^- abclist nprvmulticols newcommanddber[]ensuremathD_x R setminus qty# newcommandxloes[]x # && L_x left # right newcommandloes[][]&&L_# qty# abclist abc dber, al dfracx dfracx- &&|cdot xx- x- x &&|-x - x &&|: xloes-frac abc dber- al dfracx+x+ dfracx+ &&|cdot x+ x + &&|- x - &&|: xloes-frac abc dber al dfracx+x- - dfracx-x- dfracx-x- &&|cdot x- x+ - x- x - &&|text TU -x + x- &&|+x, + x &&|: xloes abc dber- al dfracx-x+ - dfracx-x+ dfracx-x+ &&|cdot x+ x- - x- x- &&|text TU - x- && |+ - x &&|: xloes-frac abc dber-, al dfracx+ - dfracx- dfracx+-x^ &&|cdot x+x- x- - x+ -x+ && |text TU - -x- && |+x, + xloes abc dber, al dfracxx- - dfracxx- dfracx- &&|cdot x-x- xx- - xx- x- &&|text TU -x x - &&|+x, + x &&|: xloesfrac abc dber-,, al dfracx- + dfracx dfracx+ &&|cdot xx-x+ xx+ + x-x+ xx- &&|text TU x^+x + x^- x^ - x &&| -x^, +, +x x &&|: xloesfrac abc dber-frac, al dfracx-x - dfracx-x + dfracx+ &&|cdot xx+ x+x- - x+x- + cdot x &&|text TU x^-x+x- -x^ + x - x + + x &&|text TU  x - &&|+, : xloesfrac abc dber-, -, -, al dfracx - dfracx+ dfracx+ - dfracx+ &&|cdot xx+x+x+ x+x+x+ - xx+x+ xx+x+ - xx+x+ &&|text TU x^ + x^ +x +x^ + x + - x^ -x^ - x x^ + x^ +x - x^ -x^ -x && |text TU x^ + x + x^ + x &&|-x^, -x -x &&|: xloes-frac abc dber-. al dfrac.x-x+. - dfrac.x+. &&|cdotx+. .x- - . x+. &&| text TU x - . x+. &&|-x, -. xloes-.-frac abc dber- al dfracx^+x +x+ x- &&|cdotx+ x^+x + x-x+ &&|text TU x^ +x + x^ + x - &&|-x^, -x, - x - &&|: x - && L_x , abc dber-, al dfracx^x^- - dfracx-x+ dfrac-x-x^ &&|cdotx^- x^ - x-x- --x &&|text TU x^ - x^+x - -+x && |text TU x- -+x &&| -x, + &&L_x textdber-, abc dber-, al dfracx- - dfracx+ dfracx+x^- &&|cdot x-x+ x+ - x- x + &&|text TU x + x + &&|-x,- &&L_x textdber-, abc dber-,- al dfracx-x+ - dfracx-x+ dfracx+x^+x+ &&|cdot x+x+ x+x- - x+x- x + &&|text TU x^ + x - - x^ -x + x + &&|text TU x + x + &&|-x,- -x uf:- x - &&L_x , abc Wir bestimmen den Definitionsbereich durch das Lösen der Gleichung x^-x- . Die Mitternachtsformel gibt uns die Lösungen al x_, frac pm sqrt+ frac pm x_ x_ -frac. Damit ist der Definitionsbereich dber-dfrac, . aldfracx^-x+x^-x- dfracx^+x-x^-x- uf cdotx^-x- x^-x+ x^ + x - uf -x^,-x,+ x^ -x + lf x-x- loes[x] abc Wie bei der vorherigen Teilaufgabe bestimmen wir den Definitionsbereich mit der Mitternachtsformel: al x_,  frac- pm sqrt+ frac- pm -pm &&dber-, Wir lösen die Bruchgleichung wie üblich: al dfracx^-x+x^+x- dfracx^+x+x^+x- ufcdot x^+x- x^ -x + x^ + x + uf -x^,+x,- x^ + x - lf x-x+ loes[x], abc dber-dfrac, - dfrac aldfracx-x+ dfracxx+ uf cdotx+x+ x+x- xx+ x^ +x - x^ + x uf -x^,-x x^ + x - uf : x^ + x - mf x_,  frac- pm sqrt + frac-pm   frac- pm loes[x],-frac abc dber-dfrac aldfracx + dfracx+x+ x+ uf cdot x+ xx+ + x+ x+x+ tu x^ + x + x + x^ +x + uf -x^,-x- x^ + x - mf x_, frac- pm sqrt+ frac- pm loes[x]-frac, frac abc dber aldfracx^x- - dfracx-x uf cdot x- x^+x x- uf -x,+ x^ + x + lf x+x+ loes-,- abc dber,-dfrac al. - dfracx-x+  dfracx+x uf cdot xx+ xx+ - xx- x+x+ tu x^ + x - x^ + x x^ + x + tu -x^ + x x^ + x+ uf +x^,-x x^ - x + mf x_, frac pm sqrt - frac pm loes,frac abc dber-,-, aldfracx+ + dfracx+ dfracx+ ufcdot x+x+x+ x+x+ + x+x+ x+x+ tu x^ +x + + x^ +x + x^ + x + tu x^ + x + x^+x + uf -x^,-x,- x^ -x - mf x_, frac pm sqrt+ frac pm loes-frac, abc dber-, aldfracxx- - dfracx+ dfracx- uf cdot x-x+ xx+ -x- x- tu x^ + x -x + x- tu x^ -x + x- uf-x,+ x^ -x + mf x_, frac pm sqrt- frac pm pm loes[x] abc dber-, aldfrac-x^x^- + dfrac+x^x^-x+  x tu dfrac-x^x+x- + dfrac+x^x-^ x uf cdot x+x-^ -x^x- + +x^x+ xx-^x+ tu x- -x^ +x^ +x + + x^ + x^ x^+xx^-x+ tu x^ + x x^ -x^ + x^ +x^-x^ + x uf -x^,-x x^ - x^ -x^ tu x^x-x+ loes-,, abcdber-,, aldfracx+x^-x + dfracx-x^+x dfracx+x^- tu dfracx+xx- + dfracx-xx+ dfracx+x+x- uf cdot xx-x+ x+^ + x-x- xx+ tu x^ + x + + x^ -x + x^ + x tu x^ -x + x^ + x uf -x^,-x x^ -x + lf x-x- loes abclist

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Contained in these collections:
  1. Bruchgleichungen (pw)
Tags algebra, bruchgleichungen, bruchterme, mathematik
Default Difficulty
Default points 0
Language GER
Type Calculative / Quantity