Matemáticas 2 · Tema 5
Ecuaciones con fracciones
Video
Ejemplos
Operaciones Directas con Fracciones
$$\begin{aligned}
3\left(\frac{1}{2} - \frac{4}{5}x\right) - 5x &= \frac{2}{5}x + 3 \\[14pt]
\frac{3}{2} - \frac{12}{5}x - 5x &= \frac{2}{5}x + 3 \\[14pt]
-\frac{12}{5}x - 5x - \frac{2}{5}x &= 3 - \frac{3}{2} \\[14pt]
-\frac{14}{5}x - 5x &= \frac{6 - 3}{2} \\[14pt]
\left(-\frac{14}{5} - 5\right)x &= \frac{3}{2} \\[14pt]
-\frac{39}{5}x &= \frac{3}{2} \\[14pt]
x &= \frac{3}{2} \left(\frac{5}{-39}\right) \\[14pt]
\mathbf{x = -\frac{5}{26} \approx -0.19}
\end{aligned}$$
Igualando denominador
$$\begin{aligned}
-\frac{6}{x} + \frac{4}{3x} &= \frac{7}{5} \\[14pt]
-\frac{18}{3x} + \frac{4}{3x} &= \frac{7}{5} \\[14pt]
\frac{-14}{3x} &= \frac{7}{5} \\[14pt]
-14(5) &= 7(3x) \\[14pt]
-70 &= 21x \\[14pt]
\frac{-70}{21} &= x \\[14pt]
\mathbf{x = -\frac{10}{3} \approx -3.33}
\end{aligned}$$
Multiplicación por el MCM
$$\begin{aligned}
\frac{4}{2}x - \frac{3}{5} &= -\frac{1}{4}x - \frac{4}{3} \\[14pt]
60 \left(\frac{4}{2}x - \frac{3}{5}\right) &= 60 \left(-\frac{1}{4}x - \frac{4}{3}\right) \\[14pt]
\frac{240}{2}x - \frac{180}{5} &= -\frac{60}{4}x - \frac{240}{3} \\[14pt]
120x - 36 &= -15x - 80 \\[14pt]
120x + 15x &= -80 + 36 \\[14pt]
135x &= -44 \\[14pt]
\mathbf{x = -\frac{44}{135} \approx -0.33}
\end{aligned}$$
Ejercicios
Bloque 1
$-\dfrac{3}{2x} - \dfrac{4}{6x} = \dfrac{4}{5}$
$\dfrac{3}{2}x - \dfrac{2}{5} = -\dfrac{3}{4}x - \dfrac{6}{5}$
$2\left(\dfrac{5}{6} + \dfrac{3}{8}x\right) - \dfrac{2}{5}x = 6x - 2$
Bloque 2
$-\dfrac{5}{4x} + \dfrac{2}{x} = -\dfrac{1}{3}$
$\dfrac{3}{10}x - \dfrac{1}{5} = \dfrac{4x}{3} - \dfrac{1}{2}$
$\dfrac{5}{6}x - 4 = \dfrac{7}{3} - 2\left(\dfrac{1}{3} - \dfrac{5}{4}x\right)$
Bloque 3
$-\dfrac{3}{5x} + \dfrac{2}{4x} = -\dfrac{1}{10}$
$-\dfrac{2}{3} + \dfrac{4}{5}x = -\dfrac{2}{3}x + \dfrac{5}{2}$
$-\dfrac{3}{2}x - 4\left(\dfrac{5}{3} - 3x\right) = -\dfrac{4}{2}x - \dfrac{6}{5}$
Bloque 4
$-\dfrac{6}{3x} + \dfrac{2}{4x} = -\dfrac{1}{2}$
$\dfrac{2}{6} - \dfrac{1}{2}x = \dfrac{5}{3} - \dfrac{3}{4}x$
$\dfrac{3}{4} - \dfrac{5}{2}x = \dfrac{3}{5} - 4\left(\dfrac{2}{5}x - \dfrac{1}{2}\right)$
Resoluciones
▼Bloque 1
$$\begin{aligned}
-\frac{3}{2x} - \frac{4}{6x} &= \frac{4}{5} \\[10pt]
-\frac{9}{6x} - \frac{4}{6x} &= \frac{4}{5} \\[10pt]
\frac{-13}{6x} &= \frac{4}{5} \\[10pt]
-65 &= 24x \\[10pt]
\mathbf{x = -\frac{65}{24} \approx -2.71}
\end{aligned}$$
$$\begin{aligned}
\frac{3}{2}x - \frac{2}{5} &= -\frac{3}{4}x - \frac{6}{5} \\[10pt]
20\left(\frac{3}{2}x - \frac{2}{5}\right) &= 20\left(-\frac{3}{4}x - \frac{6}{5}\right) \\[10pt]
30x - 8 &= -15x - 24 \\[10pt]
45x &= -16 \\[10pt]
\mathbf{x = -\frac{16}{45} \approx -0.36}
\end{aligned}$$
$$\begin{aligned}
2\left(\frac{5}{6} + \frac{3}{8}x\right) - \frac{2}{5}x &= 6x - 2 \\[10pt]
\frac{5}{3} + \frac{3}{4}x - \frac{2}{5}x &= 6x - 2 \\[10pt]
60 \left(\frac{5}{3} + \frac{3}{4}x - \frac{2}{5}x\right) &= 60(6x - 2) \\[10pt]
100 + 45x - 24x &= 360x - 120 \\[10pt]
100 + 21x &= 360x - 120 \\[10pt]
220 &= 339x \\[10pt]
\mathbf{x = \frac{220}{339} \approx 0.65}
\end{aligned}$$
Bloque 2
$$\begin{aligned}
-\frac{5}{4x} + \frac{2}{x} &= -\frac{1}{3} \\[10pt]
-\frac{5}{4x} + \frac{8}{4x} &= -\frac{1}{3} \\[10pt]
\frac{3}{4x} &= -\frac{1}{3} \\[10pt]
9 &= -4x \\[10pt]
\mathbf{x = -\frac{9}{4} = -2.25}
\end{aligned}$$
$$\begin{aligned}
\frac{3}{10}x - \frac{1}{5} &= \frac{4x}{3} - \frac{1}{2} \\[10pt]
30\left(\frac{3}{10}x - \frac{1}{5}\right) &= 30\left(\frac{4x}{3} - \frac{1}{2}\right) \\[10pt]
9x - 6 &= 40x - 15 \\[10pt]
9 &= 31x - 15 \\[10pt]
9 &= 31x \\[10pt]
\mathbf{x = \frac{9}{31} \approx 0.29}
\end{aligned}$$
$$\begin{aligned}
\frac{5}{6}x - 4 &= \frac{7}{3} - 2\left(\frac{1}{3} - \frac{5}{4}x\right) \\[10pt]
\frac{5}{6}x - 4 &= \frac{7}{3} - \frac{2}{3} + \frac{5}{2}x \\[10pt]
\frac{5}{6}x - 4 &= \frac{5}{3} + \frac{5}{2}x \\[10pt]
6\left(\frac{5}{6}x - 4\right) &= 6\left(\frac{5}{3} + \frac{5}{2}x\right) \\[10pt]
5x - 24 &= 10 + 15x \\[10pt]
-34 &= 10x \\[10pt]
x &= -\frac{34}{10} \\[10pt]
\mathbf{x = -3.4}
\end{aligned}$$
Bloque 3
$$\begin{aligned}
-\frac{3}{5x} + \frac{2}{4x} &= -\frac{1}{10} \\[10pt]
-\frac{6}{10x} + \frac{5}{10x} &= -\frac{1}{10} \\[10pt]
\frac{-1}{10x} &= -\frac{1}{10} \\[10pt]
-10 &= -10x \\[10pt]
\mathbf{x = 1}
\end{aligned}$$
$$\begin{aligned}
-\frac{2}{3} + \frac{4}{5}x &= -\frac{2}{3}x + \frac{5}{2} \\[10pt]
30 \left(-\frac{2}{3} + \frac{4}{5}x\right) &= 30 \left(-\frac{2}{3}x + \frac{5}{2}\right) \\[10pt]
-20 + 24x &= -20x + 75 \\[10pt]
44x &= 95 \\[10pt]
\mathbf{x = \frac{95}{44} \approx 2.16}
\end{aligned}$$
$$\begin{aligned}
-\frac{3}{2}x - 4\left(\frac{5}{3} - 3x\right) &= -\frac{4}{2}x - \frac{6}{5} \\[10pt]
-\frac{3}{2}x - \frac{20}{3} + 12x &= -2x - \frac{6}{5} \\[10pt]
30\left(-\frac{3}{2}x - \frac{20}{3} + 12x\right) &= 30\left(-2x - \frac{6}{5}\right) \\[10pt]
-45x - 200 + 360x &= -60x - 36 \\[10pt]
315x - 200 &= -60x - 36 \\[10pt]
375x &= 164 \\[10pt]
\mathbf{x = \frac{164}{375} \approx 0.44}
\end{aligned}$$
Bloque 4
$$\begin{aligned}
-\frac{6}{3x} + \frac{2}{4x} &= -\frac{1}{2} \\[10pt]
\frac{-8}{4x} + \frac{2}{4x} &= -\frac{1}{2} \\[10pt]
\frac{-6}{4x} &= -\frac{1}{2} \\[10pt]
-12 &= -4x \\[10pt]
\mathbf{x = 3}
\end{aligned}$$
$$\begin{aligned}
\frac{2}{6} - \frac{1}{2}x &= \frac{5}{3} - \frac{3}{4}x \\[10pt]
12 \left(\frac{1}{3} - \frac{1}{2}x\right) &= 12 \left(\frac{5}{3} - \frac{3}{4}x\right) \\[10pt]
4 - 6x &= 20 - 9x \\[10pt]
3x &= 16 \\[10pt]
\mathbf{x = \frac{16}{3} \approx 5.33}
\end{aligned}$$
$$\begin{aligned}
\frac{3}{4} - \frac{5}{2}x &= \frac{3}{5} - 4\left(\frac{2}{5}x - \frac{1}{2}\right) \\[10pt]
\frac{3}{4} - \frac{5}{2}x &= \frac{3}{5} - \frac{8}{5}x + 2 \\[10pt]
20\left(\frac{3}{4} - \frac{5}{2}x\right) &= 20\left(\frac{3}{5} - \frac{8}{5}x + 2\right) \\[10pt]
15 - 50x &= 12 - 32x + 40 \\[10pt]
15 - 50x &= 52 - 32x \\[10pt]
-37 &= 18x \\[10pt]
\mathbf{x = -\frac{37}{18} \approx -2.06}
\end{aligned}$$
Ejercicios extra
▼Bloque 1
$-\dfrac{4}{3x} - \dfrac{5}{6x} = \dfrac{3}{4}$
$\dfrac{5}{3}x - \dfrac{2}{7} = -\dfrac{3}{5}x - \dfrac{4}{6}$
$2\left(\dfrac{4}{5} + \dfrac{2}{7}x\right) - \dfrac{3}{5}x = 5x - 1$
Bloque 2
$-\dfrac{5}{6x} + \dfrac{3}{x} = -\dfrac{2}{5}$
$\dfrac{2}{9}x - \dfrac{3}{5} = \dfrac{3x}{4} - \dfrac{2}{5}$
$\dfrac{3}{4}x - 5 = \dfrac{6}{5} - 2\left(\dfrac{2}{3} - \dfrac{4}{5}x\right)$
Bloque 3
$-\dfrac{4}{7x} + \dfrac{3}{5x} = -\dfrac{2}{9}$
$-\dfrac{3}{5} + \dfrac{4}{6}x = -\dfrac{3}{5}x + \dfrac{2}{3}$
$-\dfrac{5}{3}x - 3\left(\dfrac{4}{3} - 2x\right) = -\dfrac{2}{4}x - \dfrac{5}{6}$
Bloque 4
$-\dfrac{2}{5x} + \dfrac{1}{2x} = -\dfrac{3}{10}$
$\dfrac{4}{3}x - \dfrac{1}{2} = \dfrac{2}{5}x + \dfrac{3}{10}$
$\dfrac{2}{3} - \dfrac{4}{5}x = \dfrac{3}{6} - 4\left(\dfrac{3}{7}x - \dfrac{2}{5}\right)$