|4x+3|<7 inequation

Given the inequality:
$$\left|{4 x + 3}\right| < 7$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{4 x + 3}\right| = 7$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$4 x + 3 \geq 0$$
or
$$- \frac{3}{4} \leq x \wedge x < \infty$$
we get the equation
$$\left(4 x + 3\right) - 7 = 0$$
after simplifying we get
$$4 x - 4 = 0$$
the solution in this interval:
$$x_{1} = 1$$

2.
$$4 x + 3 < 0$$
or
$$-\infty < x \wedge x < - \frac{3}{4}$$
we get the equation
$$\left(- 4 x - 3\right) - 7 = 0$$
after simplifying we get
$$- 4 x - 10 = 0$$
the solution in this interval:
$$x_{2} = - \frac{5}{2}$$


$$x_{1} = 1$$
$$x_{2} = - \frac{5}{2}$$
$$x_{1} = 1$$
$$x_{2} = - \frac{5}{2}$$
This roots
$$x_{2} = - \frac{5}{2}$$
$$x_{1} = 1$$
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
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{5}{2} + - \frac{1}{10}$$
=
$$- \frac{13}{5}$$
substitute to the expression
$$\left|{4 x + 3}\right| < 7$$
$$\left|{\frac{\left(-13\right) 4}{5} + 3}\right| < 7$$

37/5 < 7

but

37/5 > 7

Then
$$x < - \frac{5}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{5}{2} \wedge x < 1$$

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       x2      x1