1. **State the problem:** Simplify the expression $\left(\frac{1}{4} + \frac{1}{8} - \frac{1}{5}\right) \times \frac{4}{9}$.\n\n2. **Find a common denominator for the terms inside the parentheses:** The denominators are 4, 8, and 5. The least common denominator (LCD) is 40.\n\n3. **Rewrite each fraction with denominator 40:**\n$$\frac{1}{4} = \frac{10}{40}, \quad \frac{1}{8} = \frac{5}{40}, \quad \frac{1}{5} = \frac{8}{40}$$\n\n4. **Perform the addition and subtraction inside the parentheses:**\n$$\frac{10}{40} + \frac{5}{40} - \frac{8}{40} = \frac{10 + 5 - 8}{40} = \frac{7}{40}$$\n\n5. **Multiply the result by $\frac{4}{9}$:**\n$$\frac{7}{40} \times \frac{4}{9} = \frac{7 \times 4}{40 \times 9} = \frac{28}{360}$$\n\n6. **Simplify the fraction $\frac{28}{360}$:**\nBoth numerator and denominator are divisible by 4:\n$$\frac{\cancel{28}^{7}}{\cancel{360}^{90}} = \frac{7}{90}$$\n\n**Final answer:** $\boxed{\frac{7}{90}}$