To solve the compound inequality [tex]\(\frac{5}{2} + x \geq \frac{1}{3}\)[/tex] or [tex]\(x + 2 < -\frac{29}{6}\)[/tex], we need to solve each inequality separately and then combine the results.
1. Solving the first inequality: [tex]\(\frac{5}{2} + x \geq \frac{1}{3}\)[/tex]
Subtract [tex]\(\frac{5}{2}\)[/tex] from both sides:
[tex]\[ x \geq \frac{1}{3} - \frac{5}{2} \][/tex]
To subtract these fractions, convert them to a common denominator:
[tex]\[ \frac{1}{3} = \frac{2}{6}, \quad \frac{5}{2} = \frac{15}{6} \][/tex]
Now perform the subtraction:
[tex]\[ x \geq \frac{2}{6} - \frac{15}{6} \][/tex]
[tex]\[ x \geq \frac{2 - 15}{6} \][/tex]
[tex]\[ x \geq \frac{-13}{6} \][/tex]
So the first part of the solution is:
[tex]\[ x \geq -2.1666666666666665 \][/tex]
2. Solving the second inequality: [tex]\(x + 2 < -\frac{29}{6}\)[/tex]
Subtract 2 from both sides:
[tex]\[ x < -\frac{29}{6} - 2 \][/tex]
Convert 2 to a fraction with the same denominator:
[tex]\[ 2 = \frac{12}{6} \][/tex]
Now perform the subtraction:
[tex]\[ x < -\frac{29}{6} - \frac{12}{6} \][/tex]
[tex]\[ x < \frac{-29 - 12}{6} \][/tex]
[tex]\[ x < \frac{-41}{6} \][/tex]
So the second part of the solution is:
[tex]\[ x < -6.833333333333333 \][/tex]
Combining both parts of the solution, we have:
[tex]\[ x \geq -2.1666666666666665 \ \text{or}\ x < -6.833333333333333 \][/tex]
Comparing these solutions with the answer choices:
A. [tex]\(x \geq -\frac{2}{3}\)[/tex] or [tex]\(x < -\frac{205}{36}\)[/tex]
B. [tex]\(x \geq -\frac{13}{6}\)[/tex] or [tex]\(x < -\frac{41}{6}\)[/tex]
C. [tex]\(x \geq -\frac{2}{3}\)[/tex] or [tex]\(x \leq \frac{205}{36}\)[/tex]
D. [tex]\(x \geq -\frac{13}{6}\)[/tex] or [tex]\(x \leq \frac{41}{6}\)[/tex]
The correct choice matches:
[tex]\[ x \geq -\frac{13}{6} \ \text{or}\ x < -\frac{41}{6} \][/tex]
This corresponds to answer choice B.
So the correct answer is:
[tex]\[ \boxed{B} \][/tex]
