Question

itional details were added for this assignment.
solve the following equation for ( x ):
( \frac{9x}{4} + \frac{5}{2} = -\frac{1}{2} )
select the correct answer below:
( -\frac{4}{3} )
( -\frac{4}{9} )
( -\frac{8}{3} )
( -\frac{8}{9} )
( -\frac{10}{9} )
( -\frac{2}{9} )

Explanation:

Step1: Subtract $\frac{5}{2}$ from both sides

To isolate the term with \( x \), we subtract \(\frac{5}{2}\) from both sides of the equation \(\frac{9x}{4}+\frac{5}{2}=-\frac{1}{2}\). This gives us \(\frac{9x}{4}=-\frac{1}{2}-\frac{5}{2}\).

Step2: Simplify the right - hand side

We simplify the right - hand side by adding the fractions. Since \(-\frac{1}{2}-\frac{5}{2}=\frac{-1 - 5}{2}=\frac{-6}{2}=-3\), the equation becomes \(\frac{9x}{4}=-3\).

Step3: Solve for \( x \)

To solve for \( x \), we multiply both sides of the equation \(\frac{9x}{4}=-3\) by \(\frac{4}{9}\). So \( x=-3\times\frac{4}{9}\).
Simplifying the right - hand side, we have \( x = \frac{-12}{9}=-\frac{4}{3}\)? Wait, no, let's recalculate. Wait, \(-3\times\frac{4}{9}=\frac{-12}{9}=-\frac{4}{3}\)? Wait, no, wait:

Wait, let's go back to step 2. \(-\frac{1}{2}-\frac{5}{2}=\frac{-1 - 5}{2}=\frac{-6}{2}=-3\). Then \(\frac{9x}{4}=-3\). Multiply both sides by \(\frac{4}{9}\): \(x=-3\times\frac{4}{9}=\frac{-12}{9}=-\frac{4}{3}\)? But wait, let's check the options. Wait, maybe I made a mistake.

Wait, let's re - do the equation:

\(\frac{9x}{4}+\frac{5}{2}=-\frac{1}{2}\)

Subtract \(\frac{5}{2}\) from both sides:

\(\frac{9x}{4}=-\frac{1}{2}-\frac{5}{2}\)

\(-\frac{1}{2}-\frac{5}{2}=\frac{-1 - 5}{2}=\frac{-6}{2}=-3\)

So \(\frac{9x}{4}=-3\)

Multiply both sides by \(\frac{4}{9}\):

\(x=-3\times\frac{4}{9}=\frac{-12}{9}=-\frac{4}{3}\)

Wait, but let's check the options. The first option is \(-\frac{4}{3}\). But wait, maybe I made a mistake in calculation. Wait, let's check again:

Wait, \(\frac{9x}{4}+\frac{5}{2}=-\frac{1}{2}\)

Subtract \(\frac{5}{2}\) from both sides: \(\frac{9x}{4}=-\frac{1}{2}-\frac{5}{2}=-\frac{6}{2}=-3\)

Then \(9x=-12\) (multiplying both sides by 4)

Then \(x =-\frac{12}{9}=-\frac{4}{3}\)

Yes, so \(x =-\frac{4}{3}\)

Answer:

\(-\frac{4}{3}\) (corresponding to the first option: \(-\frac{4}{3}\))