Sure! Let's solve the inequality step-by-step:
1. Start with the given inequality:
[tex]\[
-3x + 12 \geq 7x - 8
\][/tex]
2. Move all terms involving [tex]\(x\)[/tex] to one side of the inequality and the constant terms to the other side. To do this, subtract [tex]\(7x\)[/tex] from both sides:
[tex]\[
-3x - 7x + 12 \geq -8
\][/tex]
3. Combine like terms:
[tex]\[
-10x + 12 \geq -8
\][/tex]
4. Next, move the constant term on the left side to the right side by subtracting 12 from both sides:
[tex]\[
-10x + 12 - 12 \geq -8 - 12
\][/tex]
5. Combine the constants:
[tex]\[
-10x \geq -20
\][/tex]
6. Finally, isolate [tex]\(x\)[/tex] by dividing both sides of the inequality by [tex]\(-10\)[/tex]. Here, dividing by a negative number will reverse the inequality direction:
[tex]\[
x \leq 2
\][/tex]
So, the solution to the inequality is:
[tex]\[
x \leq 2
\][/tex]
This means that all [tex]\(x\)[/tex] values less than or equal to 2 satisfy the inequality. Therefore, the solution in interval notation is:
[tex]\[
(-\infty, 2]
\][/tex]
