Question 8

We need to find the value of $x$ that makes the equation true.

Step 1 — Simplify the left side

Let's start by expanding the left side of the equation. We will use the distributive property. $\frac{5}{6}\left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{5}{6} \times \frac{3}{5}$

Now, let's multiply the fractions. $= \frac{5}{6}x + \frac{5 \times 3}{6 \times 5}$

We can simplify the product of fractions. $= \frac{5}{6}x + \frac{15}{30}$

Let's reduce the fraction $\frac{15}{30}$. $= \frac{5}{6}x + \frac{1}{2}$

$\boxed{\text{LHS} = \frac{5}{6}x + \frac{1}{2}}$

Step 2 — Compare both sides

Now we substitute the simplified left side back into the equation. The original equation was $\frac{5}{6}\left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{1}{2}$. After simplifying, the equation becomes: $\frac{5}{6}x + \frac{1}{2} = \frac{5}{6}x + \frac{1}{2}$

Let's move all terms with $x$ to one side. We will subtract $\frac{5}{6}x$ from both sides. $\frac{5}{6}x + \frac{1}{2} - \frac{5}{6}x = \frac{5}{6}x + \frac{1}{2} - \frac{5}{6}x$

This simplifies to: $\frac{1}{2} = \frac{1}{2}$

$\boxed{\text{The equation simplifies to } \frac{1}{2} = \frac{1}{2}}$

Step 3 — Determine the value of x

The simplified equation $\frac{1}{2} = \frac{1}{2}$ is always true. There is no $x$ left in the equation. This means the equation holds for any value of $x$. Since we are looking for a rational number $x$. Any rational number will satisfy this equation.

Answer

The rational number $x$ can be any rational number.