Certainly! Let's solve the inequality [tex]\(-6 - 12 \left(j - \frac{1}{2}\right) \geq 8j\)[/tex] step by step.
1. Expand the inequality:
[tex]\[
-6 - 12 \left(j - \frac{1}{2}\right) \geq 8j
\][/tex]
2. Distribute [tex]\(-12\)[/tex] within the parentheses:
[tex]\[
-6 - 12j + 6 \geq 8j
\][/tex]
3. Combine like terms on the left-hand side:
[tex]\[
-12j \geq 8j
\][/tex]
4. Move all terms involving [tex]\( j \)[/tex] to one side of the inequality:
[tex]\[
-12j - 8j \geq 0
\][/tex]
5. Combine the [tex]\( j \)[/tex] terms:
[tex]\[
-20j \geq 0
\][/tex]
6. Isolate [tex]\( j \)[/tex]:
Divide both sides by [tex]\(-20\)[/tex]. Note that dividing by a negative number reverses the inequality:
[tex]\[
j \leq 0
\][/tex]
So, the solution to the inequality is:
[tex]\[
j \leq 0
\][/tex]
Any number that is less than or equal to 0 will satisfy the inequality.
For example, [tex]\( j = -1 \)[/tex] is a solution:
[tex]\[
-6 - 12 \left( -1 - \frac{1}{2} \right) \geq 8 \cdot (-1)
\][/tex]
[tex]\[
-6 - 12 \left( -1.5 \right) \geq -8
\][/tex]
[tex]\[
-6 + 18 \geq -8
\][/tex]
[tex]\[
12 \geq -8
\][/tex]
Thus, 12 is indeed greater than or equal to -8, confirming that [tex]\( j = -1 \)[/tex] is a valid solution.
So any number that is less than or equal to 0, such as [tex]\( j = -1 \)[/tex], satisfies the inequality.
