|2x+7|−|3x+5|>0 inequation

Given the inequality:
$$\left|{2 x + 7}\right| - \left|{3 x + 5}\right| > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{2 x + 7}\right| - \left|{3 x + 5}\right| = 0$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$2 x + 7 \geq 0$$
$$3 x + 5 \geq 0$$
or
$$- \frac{5}{3} \leq x \wedge x < \infty$$
we get the equation
$$\left(2 x + 7\right) - \left(3 x + 5\right) = 0$$
after simplifying we get
$$2 - x = 0$$
the solution in this interval:
$$x_{1} = 2$$

2.
$$2 x + 7 \geq 0$$
$$3 x + 5 < 0$$
or
$$- \frac{7}{2} \leq x \wedge x < - \frac{5}{3}$$
we get the equation
$$- (- 3 x - 5) + \left(2 x + 7\right) = 0$$
after simplifying we get
$$5 x + 12 = 0$$
the solution in this interval:
$$x_{2} = - \frac{12}{5}$$

3.
$$2 x + 7 < 0$$
$$3 x + 5 \geq 0$$
The inequality system has no solutions, see the next condition

4.
$$2 x + 7 < 0$$
$$3 x + 5 < 0$$
or
$$-\infty < x \wedge x < - \frac{7}{2}$$
we get the equation
$$- (- 3 x - 5) + \left(- 2 x - 7\right) = 0$$
after simplifying we get
$$x - 2 = 0$$
the solution in this interval:
$$x_{3} = 2$$
but x3 not in the inequality interval


$$x_{1} = 2$$
$$x_{2} = - \frac{12}{5}$$
$$x_{1} = 2$$
$$x_{2} = - \frac{12}{5}$$
This roots
$$x_{2} = - \frac{12}{5}$$
$$x_{1} = 2$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{12}{5} + - \frac{1}{10}$$
=
$$- \frac{5}{2}$$
substitute to the expression
$$\left|{2 x + 7}\right| - \left|{3 x + 5}\right| > 0$$
$$- \left|{\frac{\left(-5\right) 3}{2} + 5}\right| + \left|{\frac{\left(-5\right) 2}{2} + 7}\right| > 0$$

-1/2 > 0

Then
$$x < - \frac{12}{5}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{12}{5} \wedge x < 2$$

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       x2      x1