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
]
