We are given a piecewise function for $y$ in terms of $x$ and we are asked to find the value of $x$ when $y = 25$. The function is: $y = \begin{cases} 3x - 1, & 0 \le x < 15 \\ 5x, & 15 \le x \le 25 \\ 3(x+1), & x > 25 \end{cases}$

1. Problem Description

We are given a piecewise function for $y$ in terms of $x$ and we are asked to find the value of $x$ when $y = 25$. The function is:

$y = \begin{cases}

3x - 1, & 0 \le x < 15 \\

5x, & 15 \le x \le 25 \\

3(x+1), & x > 25

\end{cases}$

2. Solution Steps

We are given that $y = 25$. We will check each case to see if we can find a corresponding value of $x$ that satisfies the given condition.

Case 1: $y = 3x - 1$ for $0 \le x < 15$

$25 = 3x - 1$

$26 = 3x$

$x = \frac{26}{3} \approx 8.67$

Since $0 \le x < 15$, this solution is valid.

Case 2: $y = 5x$ for $15 \le x \le 25$

$25 = 5x$

$x = \frac{25}{5}$

$x = 5$

Since $15 \le x \le 25$, this solution is invalid.

Case 3: $y = 3(x+1)$ for $x > 25$

$25 = 3(x+1)$

$25 = 3x + 3$

$22 = 3x$

$x = \frac{22}{3} \approx 7.33$

Since $x > 25$, this solution is invalid.

Therefore, the only valid solution is from Case 1, which gives $x = \frac{26}{3}$.

3. Final Answer

$x = \frac{26}{3}$