Type any number that is a solution to the inequality:

[tex]\[ -6 - 12\left(j - \frac{1}{2}\right) \geq 8j \][/tex]

If the inequality has no solution, type "no solution".



Answer :

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.