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absolute(x+2)=a*(x-3) la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
|x + 2| = a*(x - 3)
∣x+2∣=a(x−3)\left|{x + 2}\right| = a \left(x - 3\right)
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+2≥0x + 2 \geq 0
o
−2≤x∧x<∞-2 \leq x \wedge x < \infty
obtenemos la ecuación
−a(x−3)+(x+2)=0- a \left(x - 3\right) + \left(x + 2\right) = 0
simplificamos, obtenemos
−a(x−3)+x+2=0- a \left(x - 3\right) + x + 2 = 0
la resolución en este intervalo:
x1=3a+2a−1x_{1} = \frac{3 a + 2}{a - 1}

2.
x+2<0x + 2 < 0
o
−∞<x∧x<−2-\infty < x \wedge x < -2
obtenemos la ecuación
−a(x−3)+(−x−2)=0- a \left(x - 3\right) + \left(- x - 2\right) = 0
simplificamos, obtenemos
−a(x−3)−x−2=0- a \left(x - 3\right) - x - 2 = 0
la resolución en este intervalo:
x2=3a−2a+1x_{2} = \frac{3 a - 2}{a + 1}


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