|4/x|=8/9

To solve the equation \(\left|\frac{4}{x}\right| = \frac{8}{9}\), we can consider the two cases for the absolute value.

Case 1:

\(\frac{4}{x} = \frac{8}{9}\)

To solve for \(x\), we can multiply both sides by \(x\) (assuming \(x \neq 0\)):

\[ 4 = \frac{8}{9}x \]

Now, multiply both sides by \(9\):

\[ 36 = 8x \]

Now, divide both sides by \(8\):

\[ x = \frac{36}{8} = \frac{9}{2} = 4.5 \]

Case 2:

\(\frac{4}{x} = -\frac{8}{9}\)

Again, multiply both sides by \(x\):

\[ 4 = -\frac{8}{9}x \]

Now, multiply both sides by \(-9\):

\[ -36 = 8x \]

Now, divide both sides by \(8\):

\[ x = \frac{-36}{8} = -\frac{9}{2} = -4.5 \]

Conclusion

The solutions to the equation \(\left|\frac{4}{x}\right| = \frac{8}{9}\) are:

\[ x = \frac{9}{2} \quad \text{or} \quad x = -\frac{9}{2} \]

or, in decimal form,

\[ x = 4.5 \quad \text{or} \quad x = -4.5 \]