Solve the inequality for [tex]x[/tex].

[tex]\[ 5 - \frac{3}{2} x \geq \frac{1}{3} \][/tex]

A. [tex]x \leq \frac{28}{9}[/tex]
B. [tex]x \leq 7[/tex]
C. [tex]x \geq \frac{28}{9}[/tex]
D. [tex]x \geq 7[/tex]



Answer :

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]