|4x+2|≤6. inequation

Given the inequality:
$$\left|{4 x + 2}\right| \leq 6$$
To solve this inequality, we must first solve the corresponding equation:
$$\left|{4 x + 2}\right| = 6$$
Solve:
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.

1.
$$4 x + 2 \geq 0$$
or
$$- \frac{1}{2} \leq x \wedge x < \infty$$
we get the equation
$$\left(4 x + 2\right) - 6 = 0$$
after simplifying we get
$$4 x - 4 = 0$$
the solution in this interval:
$$x_{1} = 1$$

2.
$$4 x + 2 < 0$$
or
$$-\infty < x \wedge x < - \frac{1}{2}$$
we get the equation
$$\left(- 4 x - 2\right) - 6 = 0$$
after simplifying we get
$$- 4 x - 8 = 0$$
the solution in this interval:
$$x_{2} = -2$$


$$x_{1} = 1$$
$$x_{2} = -2$$
$$x_{1} = 1$$
$$x_{2} = -2$$
This roots
$$x_{2} = -2$$
$$x_{1} = 1$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$-2 + - \frac{1}{10}$$
=
$$- \frac{21}{10}$$
substitute to the expression
$$\left|{4 x + 2}\right| \leq 6$$
$$\left|{\frac{\left(-21\right) 4}{10} + 2}\right| \leq 6$$

32/5 <= 6

but

32/5 >= 6

Then
$$x \leq -2$$
no execute
one of the solutions of our inequality is:
$$x \geq -2 \wedge x \leq 1$$

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       x2      x1