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Solve the inequality [tex]4b + 8(2 - b) \leq 6b - 4[/tex] and write the solution in interval notation.

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GinnyAnswer
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].
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