To solve the inequality
[tex]\[
\frac{-4j - 2}{2} \leq 4j + 5,
\][/tex]
we will follow these steps:
1. Simplify the left-hand side of the inequality:
[tex]\[
\frac{-4j - 2}{2}.
\][/tex]
Divide each term in the numerator by 2:
[tex]\[
\frac{-4j}{2} + \frac{-2}{2} = -2j - 1.
\][/tex]
This simplifies the inequality to:
[tex]\[
-2j - 1 \leq 4j + 5.
\][/tex]
2. Isolate [tex]\( j \)[/tex] on one side:
To isolate [tex]\( j \)[/tex], we need to get all terms involving [tex]\( j \)[/tex] on one side and constants on the other side. Start by adding [tex]\( 2j \)[/tex] to both sides to get rid of the [tex]\( -2j \)[/tex] term on the left:
[tex]\[
-2j - 1 + 2j \leq 4j + 5 + 2j.
\][/tex]
Simplifying this, we get:
[tex]\[
-1 \leq 6j + 5.
\][/tex]
3. Move the constant term to the other side:
Subtract 5 from both sides to isolate the [tex]\( j \)[/tex]-term:
[tex]\[
-1 - 5 \leq 6j.
\][/tex]
Simplifying this, we get:
[tex]\[
-6 \leq 6j.
\][/tex]
4. Solve for [tex]\( j \)[/tex]:
Divide both sides by 6 to isolate [tex]\( j \)[/tex]:
[tex]\[
\frac{-6}{6} \leq j.
\][/tex]
Simplifying this, we get:
[tex]\[
-1 \leq j.
\][/tex]
This means [tex]\( j \)[/tex] must be greater than or equal to [tex]\(-1\)[/tex].
The solution can be written in interval notation as:
[tex]\[
j \in [-1, \infty).
\][/tex]
Thus, the solution to the inequality is
[tex]\[
\boxed{-1 \leq j < \infty}.
\][/tex]
