To solve the inequality [tex]\( 5 - \frac{3}{2} x \geq \frac{1}{3} \)[/tex], we can follow these steps:
1. Isolate the term involving [tex]\( x \)[/tex]:
We want to move the constant term [tex]\( 5 \)[/tex] to the other side of the inequality. To do this, subtract 5 from both sides:
[tex]\[
5 - \frac{3}{2} x - 5 \geq \frac{1}{3} - 5
\][/tex]
Simplifying the left side:
[tex]\[
-\frac{3}{2} x \geq \frac{1}{3} - 5
\][/tex]
2. Simplify the right side:
To combine the fractions on the right side, convert 5 into a fraction with the same denominator as [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
5 = \frac{15}{3}
\][/tex]
Now perform the subtraction:
[tex]\[
\frac{1}{3} - \frac{15}{3} = \frac{1 - 15}{3} = \frac{-14}{3}
\][/tex]
So the inequality now is:
[tex]\[
-\frac{3}{2} x \geq \frac{-14}{3}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides by [tex]\(-\frac{3}{2}\)[/tex]. Remember, when you divide or multiply both sides of an inequality by a negative number, the direction of the inequality sign reverses:
[tex]\[
x \leq \frac{\frac{-14}{3}}{-\frac{3}{2}}
\][/tex]
Simplify the right side by multiplying by the reciprocal of [tex]\(-\frac{3}{2}\)[/tex]:
[tex]\[
x \leq \frac{-14}{3} \cdot \left(-\frac{2}{3}\right)
\][/tex]
Multiplying these fractions:
[tex]\[
x \leq \frac{28}{9}
\][/tex]
So, the solution to the inequality [tex]\( 5 - \frac{3}{2} x \geq \frac{1}{3} \)[/tex] is [tex]\( x \leq \frac{28}{9} \)[/tex].
Therefore, the correct answer is:
A. [tex]\( x \leq \frac{28}{9} \)[/tex]
