Comparte esto 👍👍DESCARGACAPITULO XV Ecuaciones fraccionarias de primer grado Resolución de ecuaciones fraccionarias con denominadores compuestosEjercicio 142Resolver las siguientes ecuaciones 3 5 + 3 2x–1 =0 3(2x–1 ) +3( 5 ) 5(2x–1 ) =0 6x–3+15 =0 6x+12 =0 6x =–12 x =– 6 x =–2 2 4x–1 = 3 4x+1 2(4x+1 ) =3(4x–1 ) 8x+2 =12x–3 8x–12x =–3–2 –4x =–5 x = 5 4 5 x 2 –1 = 1 x–1 5 (x–1 )(x+1 ) = 1 x–1 5 x+1 =1 5 =x+1 5–1 =x x =4 3 x+1 – 1 x 2 –1 =0 3 x+1 = 1 x 2 –1 3 x+1 = 1 (x+1 )(x–1 ) 3 = 1 x–1 3(x–1 ) =1 3x–3 =1 3x =1+3 3x =4 x = 4 3 5x+8 3x+4 = 5x+2 3x–4 (5x+8 ) (3x–4 ) =(5x+2 ) (3x+4 ) 15 x 2 –20x+24x–32 = 15 x 2 +20x+6x+8 4x =26x+8+32 4x–26x =40 –22x =40 x =– x =– 20 11 10 x 2 –5x+8 5 x 2 +9x–19 =2 10 x 2 –5x+8 =2(5 x 2 +9x–19 ) 10 x 2 –5x+8 = 10 x 2 +18x–38 –5x–18x =–38–8 –23x =–46 x = – – 23 x =2 1 3x–3 + 1 4x+4 = 1 12x–12 1 3(x–1 ) + 1 4(x+1 ) = 1 12(x–1 ) 4(x+1 ) +3(x–1 ) 12(x–1 ) (x+1 ) = 1 12(x–1 ) 4(x+1 ) +3(x–1 ) x+1 =1 4x+4+3x–3 =x+1 7x+ 1 =x+ 1 7x–x =0 6x =0 x =0 x 4 – x 2 –8x 4x–5 = 7 4 x(4x–5 ) –4( x 2 –8x ) 4 (4x–5 ) = 7 4 4 x 2 –5x– 4 x 2 +32x =7(4x–5 ) 27x =28x–35 27x–28x =–35 –x =–35 x =35 2x–9 10 + 2x–3 2x–1 = x 5 (2x–9 ) (2x–1 ) +10(2x–3 ) (2x–1 ) = x 5 (2x–9 ) (2x–1 ) +10(2x–3 ) =2x(2x–1 ) 4 x 2 – 2x – 18x +9+ 20x –30 = 4 x 2 –2x –21 =–2x x = 21 2 x =10 1 2 (3x–1 ) 2 x–1 = 18x–1 2 2(9 x 2 –6x+1 ) =(x–1 ) (18x–1 ) 18 x 2 –12x+2 = 18 x 2 –x–18x+1 –12x =–19x+1–2 19x–12x =–1 7x =–1 x =– 1 7 2x+7 5x+2 – 2x–1 5x–4 =0 (2x+7 ) (5x–4 ) –(2x–1 ) (5x+2 ) (5x+2 ) (5x–4 ) =0 10 x 2 –8x+35x–28–(10 x 2 +4x–5x–2 ) =0 10 x 2 +27x–28– 10 x 2 +x+2 =0 28x–26 =0 28x =26 x = x = 13 14 (5x–2 ) (7x+3 ) 7x(5x–1 ) –1 =0 (5x–2 ) (7x+3 ) 7x(5x–1 ) =1 (5x–2 ) (7x+3 ) =7x(5x–1 ) 35 x 2 +15x–14x–6 = 35 x 2 –7x x+7x =6 8x =6 x = x = 3 4 3 x–4 = 2 x–3 + 8 x 2 –7x+12 3 x–4 – 2 x–3 = 8 x 2 –7x+12 3(x–3 ) –2(x–4 ) (x–4 ) (x–3 ) = 8 (x–4 ) (x–3 ) 3x–9–2x+ 8 = 8 x =9 6x–1 18 – 3(x+2 ) 5x–6 = 1+3x 9 (6x–1 ) (5x–6 ) –54(x+2 ) (5x–6 ) = 1+3x 9 30 x 2 –36x–5x+6–54x–108 2(5x–6 ) =1+3x 30 x 2 –95x–102 =2(5x–6 ) (1+3x ) 30 x 2 –95x–102 =2(5x+15 x 2 –6–18x ) 30 x 2 –95x–102 =2(15 x 2 –13x–6 ) 30 x 2 –95x–102 = 30 x 2 –26x–12 –95x+26x =102–12 –69x =90 x =– x =– 30 23 ↔ –1 7 23 5 1+x – 3 1–x – 6 1– x 2 =0 5 1+x – 3 1–x = 6 1– x 2 5(1–x ) –3(1+x ) (1–x ) (1+x ) = 6 1– x 2 5–5x–3–3x =6 2–8x =6 –8x =6–2 –8x =4 x =– 4 x =– 1 2 1+2x 1+3x – 1–2x 1–3x =– 3x–14 1–9 x 2 (1+2x ) (1–3x ) –(1–2x ) (1+3x ) (1+3x ) (1–3x ) =– 3x–14 1–9 x 2 1–3x+2x–6 x 2 –(1+3x–2x–6 x 2 ) =–3x+14 1 –x– 6 x 2 – 1 –x+ 6 x 2 =–3x+14 –2x =–3x+14 –2x+3x =14 x =14 3x–1 x 2 +7x+12 = 1 2x+6 + 7 6x+24 3x–1 (x+4 ) (x+3 ) = 1 2(x+3 ) + 7 6(x+4 ) 3x–1 (x+4 ) (x+3 ) = 3(x+4 ) +7(x+3 ) 6 (x+4 ) (x+3 ) 6(3x–1 ) =3x+12+7x+21 18x–6 =10x+33 18x–10x =33+6 8x =39 x = 39 8 ↔ 4 7 8 1 (x–1 ) 2 – 3 2x–2 =– 3 2x+2 1 (x–1 ) 2 = 3 2x–2 – 3 2x+2 1 (x–1 ) 2 = 3 2(x–1 ) – 3 2(x+1 ) 1 (x–1 ) 2 = 3(x+1 ) –3(x–1 ) 2 (x–1 )(x+1 ) 1 x–1 = 3x +3– 3x +3 2(x+1 ) 2(x+1 ) =6(x–1 ) 2x+2 =6x–6 2x–6x =–6–2 –4x =–8 x = – – 4 x =2 5x+13 15 – 4x+5 5x–15 = x 3 5x+13 15 – 4x+5 5(x–3 ) = x 3 (x–3 ) (5x+13 ) –3(4x+5 ) (x–3 ) = x 3 5 x 2 +13x–15x–39–12x–15 5(x–3 ) =x 5 x 2 –14x–54 =5x(x–3 ) 5 x 2 –14x–54 = 5 x 2 –15x 15x–14x =54 x =54 2x–1 2x+1 – x–4 3x–2 = 2 3 (2x–1 ) (3x–2 ) –(x–4 ) (2x+1 ) (2x+1 ) (3x–2 ) = 2 3 6 x 2 –4x–3x+2–(2 x 2 +x–8x–4 ) (2x+1 ) (3x–2 ) = 2 3 6 x 2 – 7x +2–2 x 2 + 7x +4 (2x+1 ) (3x–2 ) = 2 3 4 x 2 +6 (2x+1 ) (3x–2 ) = 2 3 3. 2 (2 x 2 +3 ) = 2 (2x+1 ) (3x–2 ) 6 x 2 +9 = 6 x 2 –4x+3x–2 9+2 =–x 11 =–x x =–11 4x+3 2x–5 – 3x+8 3x–7 =1 (4x+3 ) (3x–7 ) –(3x+8 ) (2x–5 ) (2x–5 ) (3x–7 ) =1 12 x 2 –28x+9x–21–(6 x 2 –15x+16x–40 ) =(2x–5 ) (3x–7 ) 12 x 2 –19x–21–6 x 2 –x+40 =6 x 2 –14x–15x+35 6 x 2 –20x+19 = 6 x 2 –29x+35 29x–20x =35–19 9x =16 x = 16 9 ↔ 1 7 9 10x–7 15x+3 = 3x+8 12 – 5 x 2 –4 20x+4 10x–7 3(5x+1 ) = 3x+8 12 – 5 x 2 –4 4(5x+1 ) 10x–7 3 (5x+1 ) = (5x+1 ) (3x+8 ) –3(5 x 2 –4 ) (5x+1 ) 4(10x–7 ) = 15 x 2 +40x+3x+8– 15 x 2 +12 40x–28 =43x+20 40x–43x =20+28 –3x =48 x =– 3 x =–16 4x–1 5 + x–2 2x–7 = 8x–3 10 –1 3 10 (4x–1 ) (2x–7 ) +5(x–2 ) 5(2x–7 ) = 8x–3 10 – 13 10 8 x 2 –28x–2x+7+5x–10 5 (2x–7 ) = 8x–3–13 8 x 2 –25x–3 2x–7 = 8x–16 2 8 x 2 –25x–3 2x–7 = (x–2 ) 2 8 x 2 –25x–3 =4(x–2 ) (2x–7 ) 8 x 2 –25x–3 =4(2 x 2 –7x–4x+14 ) 8 x 2 –25x–3 =4(2 x 2 –11x+14 ) 8 x 2 –25x–3 = 8 x 2 –44x+56 44x–25x =56+3 19x =59 x = 59 19 ↔ 3 2 19 1 x–1 – 2 x–2 = 3 2x–2 – 2 1 3 2x–4 x–2–2(x–1 ) (x–1 ) (x–2 ) = 3 2(x–1 ) – 7 3 2(x–2 ) x– 2 –2x+ 2 (x–1 ) (x–2 ) = 3 2(x–1 ) – 7 6(x–2 ) –x (x–1 ) (x–2 ) = 9(x–2 ) –7(x–1 ) 6 (x–1 ) (x–2 ) –6x =9x–18–7x+7 –6x =2x–11 –6x–2x =–11 –8x =–11 x = 11 8 ↔ 1 3 8 1 x+3 – 2 5x–20 = 1 1 2 3x–12 – 2 x+3 1 x+3 – 2 5(x–4 ) = 3 2 3 (x–4 ) – 2 x+3 5(x–4 ) –2(x+3 ) 5(x+3 ) (x–4 ) = 1 2(x–4 ) – 2 x+3 5x–20–2x–6 5 (x–4 ) (x+3 ) = x+3–4(x–4 ) 2 (x–4 ) (x+3 ) 3x–26 5 = x+3–4x+16 2 2(3x–26 ) =5(–3x+19 ) 6x–52 =–15x+95 6x+15x =95+52 21x =147 x = 21 x =7 1 6–2x – 4 5–5x = 10 12–4x – 3 10–10x 1 2(3–x ) – 4 5(1–x ) = (3–x ) – 3 10(1–x ) 1 2(3–x ) – 4 5(1–x ) = 5 2(3–x ) – 3 10(1–x ) 5(1–x ) –8(3–x ) 10(3–x ) (1–x ) = 25(1–x ) –3(3–x ) 10(3–x ) (1–x ) 5(1–x ) –8(3–x ) =25(1–x ) –3(3–x ) 5(1–x ) –25(1–x ) =8(3–x ) –3(3–x ) (1–x ) (5–25 ) =(3–x ) (8–3 ) –20(1–x ) =5(3–x ) –20+20x =15–5x 20x+5x =15+20 25x =35 x = x = 7 5 ↔ 1 2 5 2 3 – 6 x 2 9 x 2 –1 = 2 3x–1 2 3 – 6 x 2 (3x–1 ) (3x+1 ) = 2 3x–1 2(3x–1 ) (3x+1 ) –18 x 2 3(3x+1 )(3x–1 ) = 2 3x–1 2(9 x 2 –1 ) –18 x 2 3(3x+1 ) =2 18 x 2 –2– 18 x 2 =6(3x+1 ) –2 =18x+6 –6–2 =18x –8 =18x x =– x =– 4 9 5 x 2 –27x 5x+3 – 1 x =x–6 x(5 x 2 –27x ) –(5x+3 ) x(5x+3 ) =x–6 5 x 3 –27 x 2 –5x–3 =x(5x+3 ) (x–6 ) 5 x 3 –27 x 2 –5x–3 =x(5 x 2 –30x+3x–18 ) 5 x 3 – 27 x 2 –5x–3 = 5 x 3 – 27 x 2 –18x 18x–5x =3 13x =3 x = 3 13 4x+1 4x–1 – 6 16 x 2 –1 = 4x–1 4x+1 4x+1 4x–1 – 6 (4x–1 ) (4x+1 ) = 4x–1 4x+1 (4x+1 ) 2 –6 (4x+1 )(4x–1 ) = 4x–1 4x+1 16 x 2 +8x+1–6 = (4x–1 ) 2 16 x 2 +8x–5 = 16 x 2 –8x+1 8x+8x =1+5 16x =6 x = x = 3 8 3( x–1 x+1 ) +2( x+1 x–4 ) = 5x(x–1 ) x 2 –3x–4 3(x–1 ) (x–4 ) +2 (x+1 ) 2 (x+1 ) (x–4 ) = 5x(x–1 ) (x+1 ) (x–4 ) 3( x 2 –5x+4 ) +2( x 2 +2x+1 ) =5 x 2 –5x 3 x 2 –15x+12+2 x 2 +4x+2 =5 x 2 –5x 5 x 2 –11x+14 = 5 x 2 –5x –11x+5x =–14 –6x =–14 x = – – x = 7 3 ↔ 2 1 3 2( x+2 x–2 ) –3( x–2 2x+3 ) = x 2 +78 2 x 2 –x–6 2(x+2 ) (2x+3 ) –3 (x–2 ) 2 (x–2 ) (2x+3 ) = x 2 +78 2 x 2 –4x+3x–6 2(2 x 2 +3x+4x+6 ) –3( x 2 –4x+4 ) (x–2 ) (2x+3 ) = x 2 +78 2x(x–2 ) +3(x–2 ) 4 x 2 +14x+ 12 –3 x 2 +12x– 12 (x–2 ) (2x+3 ) = x 2 +78 (x–2 ) (2x+3 ) x 2 +26x = x 2 +78 x = 26 x =3 1 x 2 +3x–28 – 1 x 2 +12x+35 = 3 x 2 +x–20 1 (x+7 ) (x–4 ) – 1 (x+7 ) (x+5 ) = 3 (x+5 ) (x–4 ) x +5– x +4 (x+7 ) (x+5 ) (x–4 ) = 3 (x+5 ) (x–4 ) 9 =3(x+7 ) 9 =3x+21 9–21 =3x –12 =3x x =– 3 x =–4 x–2 x 2 +8x+7 = 2x–5 x 2 –49 – x–2 x 2 –6x–7 x–2 (x+7 ) (x+1 ) = 2x–5 (x+7 ) (x–7 ) – x–2 (x–7 ) (x+1 ) x–2 (x+7 ) (x+1 ) = (2x–5 ) (x+1 ) –(x–2 ) (x+7 ) (x–7 ) (x+7 ) (x+1 ) (x–2 ) (x–7 ) =2 x 2 +2x–5x–5–( x 2 +5x–14 ) x 2 –9x+ 14 =2 x 2 –3x–5– x 2 –5x+ 14 x 2 –9x = x 2 –8x–5 –9x+8x =–5 –x =–5 x =5 4x+5 15 x 2 +7x–2 – 2x+3 12 x 2 –7x–10 – 2x–5 20 x 2 –29x+5 =0 4x+5 15 x 2 –3x+10x–2 – 2x+3 12 x 2 –15x+8x–10 – 2x–5 20 x 2 –4x–25x+5 =0 4x+5 3x(5x–1 ) +2(5x–1 ) – 2x+3 3x(4x–5 ) +2(4x–5 ) – 2x–5 4x(5x–1 ) –5(5x–1 ) =0 4x+5 (3x+2 ) (5x–1 ) – 2x+3 (3x+2 ) (4x–5 ) – 2x–5 (4x–5 ) (5x–1 ) =0 (4x+5 ) (4x–5 ) –(2x+3 ) (5x–1 ) –(2x–5 ) (3x+2 ) (3x+2 ) (5x–1 ) (4x–5 ) =0 16 x 2 –25–(10 x 2 –2x+15x–3 ) –(6 x 2 +4x–15x–10 ) =0 16 x 2 –25–(10 x 2 +13x–3 ) –(6 x 2 –11x–10 ) =0 16 x 2 –25– 10 x 2 –13x+3– 6 x 2 +11x+10 =0 –2x–12 =0 –2x =12 x =– 2 x =–6 7 2x+1 – 3 x+4 = 2 x+1 – 3(x+1 ) 2 x 2 +9x+4 7(x+4 ) –3(2x+1 ) (2x+1 ) (x+4 ) = 2 x+1 – 3(x+1 ) 2 x 2 +x+8x+4 7x+28–6x–3 (2x+1 ) (x+4 ) = 2 x+1 – 3(x+1 ) x(2x+1 ) +4(2x+1 ) x+25 (2x+1 ) (x+4 ) = 2 x+1 – 3 (x+1 ) 2 (2x+1 ) (x+4 ) x+25 (2x+1 ) (x+4 ) = 2(2x+1 ) (x+4 ) –3( x 2 +2x+1 ) (x+1 ) (2x+1 ) (x+4 ) (x+25 ) (x+1 ) =2(2 x 2 +8x+x+4 ) –3 x 2 –6x–3 x 2 +26x+25 =4 x 2 +18x+8–3 x 2 –6x–3 x 2 +26x+25 = x 2 +12x+5 26x–12x =–25+5 14x =–20 x =– x =– 10 7 ↔ –1 3 7 (x+3 ) 2 (x–3 ) 2 = x–1 x+1 + 2(7x+1 ) x 2 –2x–3 (x+3 ) 2 (x–3 ) 2 = x–1 x+1 + 2(7x+1 ) (x–3 ) (x+1 ) (x+3 ) 2 (x–3 ) 2 = (x–1 ) (x–3 ) +2(7x+1 ) (x–3 )(x+1 ) x 2 +6x+9 x–3 = x 2 –4x+3+14x+2 x+1 x 2 +6x+9 x–3 = x 2 +10x+5 x+1 ( x 2 +6x+9 ) (x+1 ) =(x–3 ) ( x 2 +10x+5 ) x 3 +6 x 2 +9x+ x 2 +6x+9 = x 3 +10 x 2 +5x–3 x 2 –30x–15 7 x 2 +15x+9 = 7 x 2 –25x–15 15x+25x =–15–9 40x =–24 x =– x =– 3 5 x–4 x+5 – x+1 x–2 =– 12(x+3 ) (x+5 ) 2 (x–4 ) (x–2 ) –(x+1 ) (x+5 ) (x+5 )(x–2 ) =– 12(x+3 ) (x+5 ) 2 x 2 –6x+8–( x 2 +6x+5 ) x–2 =– 12(x+3 ) x+5 x 2 –6x+8– x 2 –6x–5 x–2 =– 12(x+3 ) x+5 –12x+3 x–2 =– 12(x+3 ) x+5 (–12x+3 ) (x+5 ) =(–12x–36 ) (x–2 ) –12 x 2 –60x+3x+15 = –12 x 2 +24x–36x+72 –57x =–12x+72–15 –57x+12x =57 –45x =57 x =– x =– 19 15 ↔ –1 4 15 x–3 x–4 – x–2 x–3 = x+2 x+1 – x+3 x+2 (x–3 ) 2 –(x–2 ) (x–4 ) (x–4 ) (x–3 ) = (x+2 ) 2 –(x+3 ) (x+1 ) (x+1 ) (x+2 ) x 2 –6x+9–( x 2 –6x+8 ) (x–4 ) (x–3 ) = x 2 +4x+4–( x 2 +4x+3 ) (x+1 ) (x+2 ) x 2 – 6x +9– x 2 + 6x –8 (x–4 ) (x–3 ) = x 2 + 4x +4– x 2 – 4x –3 (x+1 ) (x+2 ) 1 x 2 –7x+12 = 1 x 2 +3x+2 x 2 +3x+2 = x 2 –7x+12 3x+7x =12–2 10x =10 x = 10 10 x =1 x+6 x+2 – x+1 x–3 = x–5 x–1 – x x+4 (x+6 ) (x–3 ) –(x+1 ) (x+2 ) (x+2 ) (x–3 ) = (x–5 ) (x+4 ) –x(x–1 ) (x–1 ) (x+4 ) x 2 +3x–18–( x 2 +3x+2 ) (x+2 ) (x–3 ) = x 2 –x–20–( x 2 –x ) (x–1 ) (x+4 ) x 2 + 3x –18– x 2 – 3x –2 x 2 –x–6 = x 2 – x –20– x 2 + x x 2 +3x–4 –20 x 2 –x–6 = –20 x 2 +3x–4 1 x 2 –x–6 = 1 x 2 +3x–4 x 2 +3x–4 = x 2 –x–6 3x+x =–6+4 4x =–2 x =– 2 x =– 1 2 Categories: Capítulo XV