(abs(x-(7/2)))+(23/10)<=(2583/1000) inequation

Given the inequality:
$$\left|{x - \frac{7}{2}}\right| + \frac{23}{10} \leq \frac{2583}{1000}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{x - \frac{7}{2}}\right| + \frac{23}{10} = \frac{2583}{1000}$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$x - \frac{7}{2} \geq 0$$
or
$$\frac{7}{2} \leq x \wedge x < \infty$$
we get the equation
$$\left(x - \frac{7}{2}\right) - \frac{283}{1000} = 0$$
after simplifying we get
$$x - \frac{3783}{1000} = 0$$
the solution in this interval:
$$x_{1} = \frac{3783}{1000}$$

2.
$$x - \frac{7}{2} < 0$$
or
$$-\infty < x \wedge x < \frac{7}{2}$$
we get the equation
$$\left(\frac{7}{2} - x\right) - \frac{283}{1000} = 0$$
after simplifying we get
$$\frac{3217}{1000} - x = 0$$
the solution in this interval:
$$x_{2} = \frac{3217}{1000}$$


$$x_{1} = \frac{3783}{1000}$$
$$x_{2} = \frac{3217}{1000}$$
$$x_{1} = \frac{3783}{1000}$$
$$x_{2} = \frac{3217}{1000}$$
This roots
$$x_{2} = \frac{3217}{1000}$$
$$x_{1} = \frac{3783}{1000}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{3217}{1000}$$
=
$$\frac{3117}{1000}$$
substitute to the expression
$$\left|{x - \frac{7}{2}}\right| + \frac{23}{10} \leq \frac{2583}{1000}$$
$$\left|{- \frac{7}{2} + \frac{3117}{1000}}\right| + \frac{23}{10} \leq \frac{2583}{1000}$$

2683    2583
---- <= ----
1000    1000

but

2683    2583
---- >= ----
1000    1000

Then
$$x \leq \frac{3217}{1000}$$
no execute
one of the solutions of our inequality is:
$$x \geq \frac{3217}{1000} \wedge x \leq \frac{3783}{1000}$$

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        /     \  
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       x2      x1