absolute(5*x+8)<17 inequation

Given the inequality:
$$\left|{5 x + 8}\right| < 17$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{5 x + 8}\right| = 17$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$5 x + 8 \geq 0$$
or
$$- \frac{8}{5} \leq x \wedge x < \infty$$
we get the equation
$$\left(5 x + 8\right) - 17 = 0$$
after simplifying we get
$$5 x - 9 = 0$$
the solution in this interval:
$$x_{1} = \frac{9}{5}$$

2.
$$5 x + 8 < 0$$
or
$$-\infty < x \wedge x < - \frac{8}{5}$$
we get the equation
$$\left(- 5 x - 8\right) - 17 = 0$$
after simplifying we get
$$- 5 x - 25 = 0$$
the solution in this interval:
$$x_{2} = -5$$


$$x_{1} = \frac{9}{5}$$
$$x_{2} = -5$$
$$x_{1} = \frac{9}{5}$$
$$x_{2} = -5$$
This roots
$$x_{2} = -5$$
$$x_{1} = \frac{9}{5}$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-5 + - \frac{1}{10}$$
=
$$- \frac{51}{10}$$
substitute to the expression
$$\left|{5 x + 8}\right| < 17$$
$$\left|{\frac{\left(-51\right) 5}{10} + 8}\right| < 17$$

35/2 < 17

but

35/2 > 17

Then
$$x < -5$$
no execute
one of the solutions of our inequality is:
$$x > -5 \wedge x < \frac{9}{5}$$

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