The problem asks us to simplify the expression $\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8}$ and express the result in the form $px+q$, where $p$ and $q$ are rational numbers. Then we need to state the values of $p$ and $q$.

1. Problem Description

The problem asks us to simplify the expression $\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8}$ and express the result in the form $px+q$, where $p$ and $q$ are rational numbers. Then we need to state the values of $p$ and $q$.

2. Solution Steps

First, we find a common denominator for the fractions, which is

2

4. We rewrite each fraction with the common denominator:

$\frac{2x-1}{3} = \frac{8(2x-1)}{24} = \frac{16x - 8}{24}$

$\frac{x-4}{4} = \frac{6(x-4)}{24} = \frac{6x - 24}{24}$

$\frac{2x+1}{8} = \frac{3(2x+1)}{24} = \frac{6x + 3}{24}$

Now, we substitute these equivalent fractions into the original expression:

$\frac{2x-1}{3} - \frac{x-4}{4} + \frac{2x+1}{8} = \frac{16x - 8}{24} - \frac{6x - 24}{24} + \frac{6x + 3}{24}$

Combine the fractions:

$\frac{(16x - 8) - (6x - 24) + (6x + 3)}{24}$

Distribute the negative sign and combine like terms:

$\frac{16x - 8 - 6x + 24 + 6x + 3}{24} = \frac{(16x - 6x + 6x) + (-8 + 24 + 3)}{24}$

$\frac{16x + 19}{24}$

Separate the terms:

$\frac{16x}{24} + \frac{19}{24}$

Simplify the fraction $\frac{16}{24}$:

$\frac{16}{24} = \frac{2}{3}$

Thus, we have:

$\frac{2}{3}x + \frac{19}{24}$

So, we have the expression in the form $px + q$ with $p = \frac{2}{3}$ and $q = \frac{19}{24}$.

3. Final Answer

$p = \frac{2}{3}$

$q = \frac{19}{24}$