To solve the inequality [tex]\(4b + 8(2 - b) \leq 6b - 4\)[/tex], let's go step-by-step.
1. Expand and simplify both sides of the inequality:
[tex]\[4b + 8(2 - b) \leq 6b - 4\][/tex]
Expand [tex]\(8(2 - b)\)[/tex]:
[tex]\[4b + 16 - 8b \leq 6b - 4\][/tex]
2. Combine like terms on the left-hand side:
[tex]\[4b - 8b + 16 \leq 6b - 4\][/tex]
[tex]\[ -4b + 16 \leq 6b - 4\][/tex]
3. Let's isolate the variable [tex]\(b\)[/tex] on one side of the inequality:
Add [tex]\(4b\)[/tex] to both sides to get all terms involving [tex]\(b\)[/tex] on the right-hand side:
[tex]\[16 \leq 10b - 4\][/tex]
4. Next, isolate the constant term on one side:
Add [tex]\(4\)[/tex] to both sides:
[tex]\[16 + 4 \leq 10b\][/tex]
[tex]\[20 \leq 10b\][/tex]
5. Now solve for [tex]\(b\)[/tex]:
Divide both sides by [tex]\(10\)[/tex]:
[tex]\[2 \leq b\][/tex]
or
[tex]\[b \geq 2\][/tex]
6. Write the solution set in interval notation:
The inequality [tex]\(b \geq 2\)[/tex] can be written in interval notation as:
[tex]\[[2, \infty)\][/tex]
So, the solution to the inequality [tex]\(4b + 8(2 - b) \leq 6b - 4\)[/tex] is [tex]\(b \in [2, \infty)\)[/tex].
