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|x-2|+|x-4|=3

|x-2|+|x-4|=3 equation

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Numerical solution:

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The solution

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|x - 2| + |x - 4| = 3
∣x−4∣+∣x−2∣=3\left|{x - 4}\right| + \left|{x - 2}\right| = 3
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−2≥0x - 2 \geq 0
or
4≤x∧x<∞4 \leq x \wedge x < \infty
we get the equation
(x−4)+(x−2)−3=0\left(x - 4\right) + \left(x - 2\right) - 3 = 0
after simplifying we get
2x−9=02 x - 9 = 0
the solution in this interval:
x1=92x_{1} = \frac{9}{2}

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

3.
x−4<0x - 4 < 0
x−2≥0x - 2 \geq 0
or
2≤x∧x<42 \leq x \wedge x < 4
we get the equation
(4−x)+(x−2)−3=0\left(4 - x\right) + \left(x - 2\right) - 3 = 0
after simplifying we get
incorrect
the solution in this interval:

4.
x−4<0x - 4 < 0
x−2<0x - 2 < 0
or
−∞<x∧x<2-\infty < x \wedge x < 2
we get the equation
(2−x)+(4−x)−3=0\left(2 - x\right) + \left(4 - x\right) - 3 = 0
after simplifying we get
3−2x=03 - 2 x = 0
the solution in this interval:
x2=32x_{2} = \frac{3}{2}


The final answer:
x1=92x_{1} = \frac{9}{2}
x2=32x_{2} = \frac{3}{2}
The graph
05-10-5101520025
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
x1 = 3/2
x1=32x_{1} = \frac{3}{2} 
x2 = 9/2
x2=92x_{2} = \frac{9}{2} 
x2 = 9/2
Sum and product of roots [src] 
sum
3/2 + 9/2
32+92\frac{3}{2} + \frac{9}{2} 
=
6
66 
product
3*9
---
2*2
3⋅92⋅2\frac{3 \cdot 9}{2 \cdot 2} 
=
27/4
274\frac{27}{4} 
27/4
Numerical answer [src] 
x1 = 4.5
x2 = 1.5
x2 = 1.5
The graph
|x-2|+|x-4|=3 equation
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