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To solve the equation \( 3m + \frac{5}{4} - \frac{3m}{4} = -2 \), we start by simplifying the equation.
First, let's rewrite the left side:
\[
3m - \frac{3m}{4} + \frac{5}{4}
\]
To combine the terms involving \( m \), we can express \( 3m \) as a fraction with a denominator of 4:
\[
3m = \frac{12m}{4}
\]
Now substitute that back into the equation:
\[
\frac{12m}{4} - \frac{3m}{4} + \frac{5}{4} = -2
\]
Now combine the terms:
\[
\frac{12m - 3m + 5}{4} = -2
\]
\[
\frac{9m + 5}{4} = -2
\]
Next, we eliminate the fraction by multiplying both sides of the equation by 4:
\[
9m + 5 = -8
\]
Now, isolate \( m \) by subtracting 5 from both sides:
\[
9m = -8 - 5
\]
\[
9m = -13
\]
Finally, divide both sides by 9:
\[
m = -\frac{13}{9}
\]
Thus, the value for \( m \) that makes the equation true is:
\[
m = -\frac{13}{9}
\]