Solve for [tex]$x$[/tex].

[tex]$x + 4 \leq -2$[/tex] and [tex][tex]$3x - 2 \geq 1$[/tex][/tex]

A. No solution
B. Infinite number of solutions
C. Submit
D. Pass



Answer :

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