Sure, let's solve the inequality step-by-step:
[tex]\[ -3(k + 8) \geq 7(k - 12) \][/tex]
1. First, distribute [tex]\(-3\)[/tex] on the left side and [tex]\(7\)[/tex] on the right side:
[tex]\[ -3k - 24 \geq 7k - 84 \][/tex]
2. Next, let's move all terms involving [tex]\(k\)[/tex] to one side and constant terms to the other side. Start by adding [tex]\(3k\)[/tex] to both sides:
[tex]\[ -24 \geq 10k - 84 \][/tex]
3. Then add [tex]\(84\)[/tex] to both sides to isolate the [tex]\(10k\)[/tex] term:
[tex]\[ 60 \geq 10k \][/tex]
4. Finally, divide both sides by [tex]\(10\)[/tex] to solve for [tex]\(k\)[/tex]:
[tex]\[ 6 \geq k \][/tex]
Alternatively written:
[tex]\[ k \leq 6 \][/tex]
So, the solution to the inequality is:
[tex]\[ k \leq 6 \][/tex]
This can be interpreted as:
[tex]\[ -\infty < k \leq 6 \][/tex]
Thus, [tex]\(k\)[/tex] can be any real number less than or equal to [tex]\(6\)[/tex].
