Let's solve each inequality step-by-step and then combine the results to determine the solution for [tex]\( x \)[/tex].
1. Solve the first inequality [tex]\( x + 4 \leq -2 \)[/tex]:
- Start with the inequality: [tex]\( x + 4 \leq -2 \)[/tex].
- Subtract 4 from both sides to isolate [tex]\( x \)[/tex]:
[tex]\[
x + 4 - 4 \leq -2 - 4
\][/tex]
[tex]\[
x \leq -6
\][/tex]
2. Solve the second inequality [tex]\( 3x - 2 \geq 1 \)[/tex]:
- Start with the inequality: [tex]\( 3x - 2 \geq 1 \)[/tex].
- Add 2 to both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
3x - 2 + 2 \geq 1 + 2
\][/tex]
[tex]\[
3x \geq 3
\][/tex]
- Now, divide both sides by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{3x}{3} \geq \frac{3}{3}
\][/tex]
[tex]\[
x \geq 1
\][/tex]
Now, let's combine the results of these inequalities:
- From the first inequality, we have [tex]\( x \leq -6 \)[/tex].
- From the second inequality, we have [tex]\( x \geq 1 \)[/tex].
For [tex]\( x \)[/tex] to satisfy both inequalities simultaneously, it must be less than or equal to [tex]\(-6\)[/tex] and greater than or equal to [tex]\(1\)[/tex] at the same time. However, there is no number [tex]\( x \)[/tex] that can satisfy both conditions simultaneously. Therefore, there is no solution for the given system of inequalities.
In conclusion, the solution is:
- No solution
