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ax-|x|-|x+2|=a/2 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                      a
a*x - |x| - |x + 2| = -
                      2
$$\left(a x - \left|{x}\right|\right) - \left|{x + 2}\right| = \frac{a}{2}$$
Simplificar
Solución detallada
Para cada expresión dentro del módulo en la ecuación
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.

1.
$$x \geq 0$$
$$x + 2 \geq 0$$
o
$$0 \leq x \wedge x < \infty$$
obtenemos la ecuación
$$a x - \frac{a}{2} - x - \left(x + 2\right) = 0$$
simplificamos, obtenemos
$$a x - \frac{a}{2} - 2 x - 2 = 0$$
la resolución en este intervalo:
$$x_{1} = \frac{a + 4}{2 \left(a - 2\right)}$$

2.
$$x \geq 0$$
$$x + 2 < 0$$
Las desigualdades no se cumplen, hacemos caso omiso

3.
$$x < 0$$
$$x + 2 \geq 0$$
o
$$-2 \leq x \wedge x < 0$$
obtenemos la ecuación
$$a x - \frac{a}{2} - - x - \left(x + 2\right) = 0$$
simplificamos, obtenemos
$$a x - \frac{a}{2} - 2 = 0$$
la resolución en este intervalo:
$$x_{2} = \frac{a + 4}{2 a}$$

4.
$$x < 0$$
$$x + 2 < 0$$
o
$$-\infty < x \wedge x < -2$$
obtenemos la ecuación
$$a x - \frac{a}{2} - - x - \left(- x - 2\right) = 0$$
simplificamos, obtenemos
$$a x - \frac{a}{2} + 2 x + 2 = 0$$
la resolución en este intervalo:
$$x_{3} = \frac{a - 4}{2 \left(a + 2\right)}$$


Entonces la respuesta definitiva es:
$$x_{1} = \frac{a + 4}{2 \left(a - 2\right)}$$
$$x_{2} = \frac{a + 4}{2 a}$$
$$x_{3} = \frac{a - 4}{2 \left(a + 2\right)}$$
Gráfica
Respuesta rápida [src] 
         //4 + a                            \     //4 + a                            \
         ||-----  for And(a <= -4/5, a > -4)|     ||-----  for And(a <= -4/5, a > -4)|
x1 = I*im|< 2*a                             | + re|< 2*a                             |
         ||                                 |     ||                                 |
         \\ nan           otherwise         /     \\ nan           otherwise         /
$$x_{1} = \operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
         //  -4 + a                            \     //  -4 + a                            \
         ||---------  for And(a > -2, a < -4/5)|     ||---------  for And(a > -2, a < -4/5)|
x2 = I*im|<2*(2 + a)                           | + re|<2*(2 + a)                           |
         ||                                    |     ||                                    |
         \\   nan             otherwise        /     \\   nan             otherwise        /
$$x_{2} = \operatorname{re}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
         //  4 + a           4 + a        \     //  4 + a           4 + a        \
         ||----------  for ---------- >= 0|     ||----------  for ---------- >= 0|
x3 = I*im|<2*(-2 + a)      2*(-2 + a)     | + re|<2*(-2 + a)      2*(-2 + a)     |
         ||                               |     ||                               |
         \\   nan           otherwise     /     \\   nan           otherwise     /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
x3 = re(Piecewise(((a + 4)/(2*(a - 2), (a + 4)/(2*(a - 2)) >= 0), (nan, True))) + i*im(Piecewise(((a + 4)/(2*(a - 2)), (a + 4)/(2*(a - 2)) >= 0), (nan, True))))
Suma y producto de raíces [src] 
suma
    //4 + a                            \     //4 + a                            \       //  -4 + a                            \     //  -4 + a                            \       //  4 + a           4 + a        \     //  4 + a           4 + a        \
    ||-----  for And(a <= -4/5, a > -4)|     ||-----  for And(a <= -4/5, a > -4)|       ||---------  for And(a > -2, a < -4/5)|     ||---------  for And(a > -2, a < -4/5)|       ||----------  for ---------- >= 0|     ||----------  for ---------- >= 0|
I*im|< 2*a                             | + re|< 2*a                             | + I*im|<2*(2 + a)                           | + re|<2*(2 + a)                           | + I*im|<2*(-2 + a)      2*(-2 + a)     | + re|<2*(-2 + a)      2*(-2 + a)     |
    ||                                 |     ||                                 |       ||                                    |     ||                                    |       ||                               |     ||                               |
    \\ nan           otherwise         /     \\ nan           otherwise         /       \\   nan             otherwise        /     \\   nan             otherwise        /       \\   nan           otherwise     /     \\   nan           otherwise     /
$$\left(\left(\operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$ 
=
    //4 + a                            \       //  4 + a           4 + a        \       //  -4 + a                            \     //4 + a                            \     //  4 + a           4 + a        \     //  -4 + a                            \
    ||-----  for And(a <= -4/5, a > -4)|       ||----------  for ---------- >= 0|       ||---------  for And(a > -2, a < -4/5)|     ||-----  for And(a <= -4/5, a > -4)|     ||----------  for ---------- >= 0|     ||---------  for And(a > -2, a < -4/5)|
I*im|< 2*a                             | + I*im|<2*(-2 + a)      2*(-2 + a)     | + I*im|<2*(2 + a)                           | + re|< 2*a                             | + re|<2*(-2 + a)      2*(-2 + a)     | + re|<2*(2 + a)                           |
    ||                                 |       ||                               |       ||                                    |     ||                                 |     ||                               |     ||                                    |
    \\ nan           otherwise         /       \\   nan           otherwise     /       \\   nan             otherwise        /     \\ nan           otherwise         /     \\   nan           otherwise     /     \\   nan             otherwise        /
$$\operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$ 
producto
/    //4 + a                            \     //4 + a                            \\ /    //  -4 + a                            \     //  -4 + a                            \\ /    //  4 + a           4 + a        \     //  4 + a           4 + a        \\
|    ||-----  for And(a <= -4/5, a > -4)|     ||-----  for And(a <= -4/5, a > -4)|| |    ||---------  for And(a > -2, a < -4/5)|     ||---------  for And(a > -2, a < -4/5)|| |    ||----------  for ---------- >= 0|     ||----------  for ---------- >= 0||
|I*im|< 2*a                             | + re|< 2*a                             ||*|I*im|<2*(2 + a)                           | + re|<2*(2 + a)                           ||*|I*im|<2*(-2 + a)      2*(-2 + a)     | + re|<2*(-2 + a)      2*(-2 + a)     ||
|    ||                                 |     ||                                 || |    ||                                    |     ||                                    || |    ||                               |     ||                               ||
\    \\ nan           otherwise         /     \\ nan           otherwise         // \    \\   nan             otherwise        /     \\   nan             otherwise        // \    \\   nan           otherwise     /     \\   nan           otherwise     //
$$\left(\operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 a} & \text{for}\: a \leq - \frac{4}{5} \wedge a > -4 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a - 4}{2 \left(a + 2\right)} & \text{for}\: a > -2 \wedge a < - \frac{4}{5} \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a + 4}{2 \left(a - 2\right)} & \text{for}\: \frac{a + 4}{2 \left(a - 2\right)} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$ 
=
nan
$$\text{NaN}$$ 
nan
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