What is a possible value of x if 3/5 < 1/x < 7/9 ?

Step 1: Invert the inequality. Given the inequality $\frac{3}{5} < \frac{1}{x} < \frac{7}{9}$. Since all parts of the inequality are positive, we can invert each fraction. When inverting, the inequality signs must be reversed. $\frac{5}{3} > x > \frac{9}{7}$

Step 2: Rewrite the inequality in ascending order. It is standard to write inequalities with the smallest value on the left. $\frac{9}{7} < x < \frac{5}{3}$

Step 3: Convert the fractions to decimals or find a common denominator to identify a value for $x$. To find a value for $x$, we can convert the fractions to decimals: $\frac{9}{7} \approx 1.2857$ $\frac{5}{3} \approx 1.6667$ So, we are looking for a value of $x$ such that $1.2857 < x < 1.6667$.

Alternatively, find a common denominator for $\frac{9}{7}$ and $\frac{5}{3}$. The least common multiple of 7 and 3 is 21. $\frac{9}{7} = \frac{9 \times 3}{7 \times 3} = \frac{27}{21}$ $\frac{5}{3} = \frac{5 \times 7}{3 \times 7} = \frac{35}{21}$ So the inequality is: $\frac{27}{21} < x < \frac{35}{21}$

Step 4: Identify a possible value for $x$. Any fraction between $\frac{27}{21}$ and $\frac{35}{21}$ (exclusive) is a possible value for $x$. For example, $\frac{28}{21}$ is between these values. Simplifying $\frac{28}{21}$: $\frac{28}{21} = \frac{4 \times 7}{3 \times 7} = \frac{4}{3}$ Let's check this value: $\frac{9}{7} \approx 1.2857$ $\frac{4}{3} \approx 1.3333$ $\frac{5}{3} \approx 1.6667$ Since $1.2857 < 1.3333 < 1.6667$, $\frac{4}{3}$ is a valid possible value for $x$.

A possible value of $x$ is $\boxed{\frac{4}{3}}$.

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