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|x-4|+|x+4|=a equation

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Numerical solution:

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The solution

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|x - 4| + |x + 4| = a
∣x−4∣+∣x+4∣=a\left|{x - 4}\right| + \left|{x + 4}\right| = a
Simplify
Detail solution
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
x−4≥0x - 4 \geq 0
x+4≥0x + 4 \geq 0
or
4≤x∧x<∞4 \leq x \wedge x < \infty
we get the equation
−a+(x−4)+(x+4)=0- a + \left(x - 4\right) + \left(x + 4\right) = 0
after simplifying we get
−a+2x=0- a + 2 x = 0
the solution in this interval:
x1=a2x_{1} = \frac{a}{2}

2.
x−4≥0x - 4 \geq 0
x+4<0x + 4 < 0
The inequality system has no solutions, see the next condition

3.
x−4<0x - 4 < 0
x+4≥0x + 4 \geq 0
or
−4≤x∧x<4-4 \leq x \wedge x < 4
we get the equation
−a+(4−x)+(x+4)=0- a + \left(4 - x\right) + \left(x + 4\right) = 0
after simplifying we get
8−a=08 - a = 0
the solution in this interval:

4.
x−4<0x - 4 < 0
x+4<0x + 4 < 0
or
−∞<x∧x<−4-\infty < x \wedge x < -4
we get the equation
−a+(4−x)+(−x−4)=0- a + \left(4 - x\right) + \left(- x - 4\right) = 0
after simplifying we get
−a−2x=0- a - 2 x = 0
the solution in this interval:
x2=−a2x_{2} = - \frac{a}{2}


The final answer:
x1=a2x_{1} = \frac{a}{2}
x2=−a2x_{2} = - \frac{a}{2}
The graph
Rapid solution [src] 
         //-a            \     //-a            \
         ||---  for a > 8|     ||---  for a > 8|
x1 = I*im|< 2            | + re|< 2            |
         ||              |     ||              |
         \\nan  otherwise/     \\nan  otherwise/
x1=re⁡({−a2for a>8NaNotherwise)+iim⁡({−a2for a>8NaNotherwise)x_{1} = \operatorname{re}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
         // a             \     // a             \
         || -   for a >= 8|     || -   for a >= 8|
x2 = I*im|< 2             | + re|< 2             |
         ||               |     ||               |
         \\nan  otherwise /     \\nan  otherwise /
x2=re⁡({a2for a≥8NaNotherwise)+iim⁡({a2for a≥8NaNotherwise)x_{2} = \operatorname{re}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
Eq(x2, re(Piecewise((a/2, a >= 8), (nan, True))) + i*im(Piecewise((a/2, a >= 8), (nan, True))))
Sum and product of roots [src] 
sum
    //-a            \     //-a            \       // a             \     // a             \
    ||---  for a > 8|     ||---  for a > 8|       || -   for a >= 8|     || -   for a >= 8|
I*im|< 2            | + re|< 2            | + I*im|< 2             | + re|< 2             |
    ||              |     ||              |       ||               |     ||               |
    \\nan  otherwise/     \\nan  otherwise/       \\nan  otherwise /     \\nan  otherwise /
(re⁡({−a2for a>8NaNotherwise)+iim⁡({−a2for a>8NaNotherwise))+(re⁡({a2for a≥8NaNotherwise)+iim⁡({a2for a≥8NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
    // a             \       //-a            \     // a             \     //-a            \
    || -   for a >= 8|       ||---  for a > 8|     || -   for a >= 8|     ||---  for a > 8|
I*im|< 2             | + I*im|< 2            | + re|< 2             | + re|< 2            |
    ||               |       ||              |     ||               |     ||              |
    \\nan  otherwise /       \\nan  otherwise/     \\nan  otherwise /     \\nan  otherwise/
re⁡({−a2for a>8NaNotherwise)+re⁡({a2for a≥8NaNotherwise)+iim⁡({−a2for a>8NaNotherwise)+iim⁡({a2for a≥8NaNotherwise)\operatorname{re}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} 
product
/    //-a            \     //-a            \\ /    // a             \     // a             \\
|    ||---  for a > 8|     ||---  for a > 8|| |    || -   for a >= 8|     || -   for a >= 8||
|I*im|< 2            | + re|< 2            ||*|I*im|< 2             | + re|< 2             ||
|    ||              |     ||              || |    ||               |     ||               ||
\    \\nan  otherwise/     \\nan  otherwise// \    \\nan  otherwise /     \\nan  otherwise //
(re⁡({−a2for a>8NaNotherwise)+iim⁡({−a2for a>8NaNotherwise))(re⁡({a2for a≥8NaNotherwise)+iim⁡({a2for a≥8NaNotherwise))\left(\operatorname{re}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{2} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{2} & \text{for}\: a \geq 8 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) 
=
/                  2            
|-(I*im(a) + re(a))             
|--------------------  for a > 8
<         4                     
|                               
|        nan           otherwise
\
{−(re⁡(a)+iim⁡(a))24for a>8NaNotherwise\begin{cases} - \frac{\left(\operatorname{re}{\left(a\right)} + i \operatorname{im}{\left(a\right)}\right)^{2}}{4} & \text{for}\: a > 8 \\\text{NaN} & \text{otherwise} \end{cases} 
Piecewise((-(i*im(a) + re(a))^2/4, a > 8), (nan, True))
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