To determine the slope of the line that passes through the points [tex]\(\left(-3, -\frac{5}{2}\right)\)[/tex] and [tex]\((3, -8)\)[/tex], we can use the slope formula, which is:
[tex]\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\][/tex]
where [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] are coordinates of the two points.
1. Substitute the coordinates into the slope formula:
[tex]\[
m = \frac{-8 - \left(-\frac{5}{2}\right)}{3 - (-3)}
\][/tex]
2. Simplify the numerator:
[tex]\[
-8 - \left(-\frac{5}{2}\right) = -8 + \frac{5}{2}
\][/tex]
To combine these terms, convert [tex]\(-8\)[/tex] to a fraction with a denominator of 2:
[tex]\[
-8 = \frac{-16}{2}
\][/tex]
So,
[tex]\[
-8 + \frac{5}{2} = \frac{-16}{2} + \frac{5}{2} = \frac{-16 + 5}{2} = \frac{-11}{2}
\][/tex]
3. Simplify the denominator:
[tex]\[
3 - (-3) = 3 + 3 = 6
\][/tex]
4. Combine the simplified numerator and denominator:
[tex]\[
m = \frac{\frac{-11}{2}}{6}
\][/tex]
5. Simplify the fraction:
To simplify [tex]\(\frac{\frac{-11}{2}}{6}\)[/tex], we multiply the numerator by the reciprocal of the denominator:
[tex]\[
m = \frac{-11}{2} \times \frac{1}{6} = \frac{-11}{12}
\][/tex]
Therefore, the slope of the line that contains the points [tex]\(\left(-3, -\frac{5}{2}\right)\)[/tex] and [tex]\( (3, -8) \)[/tex] is:
[tex]\[
m = -\frac{11}{12}
\][/tex]
So, the correct answer is [tex]\(\boxed{-\frac{11}{12}}\)[/tex].
