The value of y for which $\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}$ is

To solve the inequality $\frac{1}{5}y + \frac{1}{5} < \frac{1}{2}y + \frac{2}{5}$, follow these steps:

1. Analyze the inequality

We have a linear inequality with fractions. To make the calculation easier, we should eliminate the fractions by multiplying the entire inequality by the Least Common Multiple (LCM) of the denominators (5, 5, 2, 5). The LCM is 10.

2. Clear the fractions

Multiply every term on both sides by 10: $10 \left( \frac{1}{5}y \right) + 10 \left( \frac{1}{5} \right) < 10 \left( \frac{1}{2}y \right) + 10 \left( \frac{2}{5} \right)$

Simplify each term: $2y + 2 < 5y + 4$

3. Solve for $y$

Group the $y$ terms on one side and the constants on the other. Let's move $2y$ to the right and $4$ to the left: $2 - 4 < 5y - 2y$

Simplify: $-2 < 3y$

Now, divide both sides by 3 to isolate $y$: $-\frac{2}{3} < y$

This can also be written as: $y > -\frac{2}{3}$

4. Conclusion

Comparing our result to the given options, we find that $y > -\frac{2}{3}$ matches option C.

Exam Tip: When working with inequalities, remember that if you multiply or divide by a negative number, you must flip the inequality sign. In this problem, we kept the coefficients positive by moving terms strategically, which helped us avoid that common pitfall!