Verified answer

[tex]( \frac{4}{9} ) {}^{x} + ( \frac{2}{3} ) {}^{x - 2} = 0 \\ [/tex]

[tex]( \frac{4}{9} ) {}^{x} = - ( \frac{2}{3} ) {}^{x - 2} \\ [/tex]

[tex] \frac{( \frac{4}{9} ) {}^{x} }{( \frac{2}{3} ) {}^{x - 2} } = - 1 \\ [/tex]

[tex]\boxed{\boxed{\boxed{x∉ℝ}}} \\ [/tex]