Let's solve the equation step by step.
The given equation is:
\[ \frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5 \]
First, let's combine like terms on both sides of the equation.
Step 1: Simplify the left side
On the left side, combine \(\frac{3}{4}x\) and \(-2x\).
To do this, we convert \(-2x\) to a fraction with a common denominator. We can write \(-2x\) as \(-\frac{8}{4}x\):
\[ \frac{3}{4}x - \frac{8}{4}x + 3 = -\frac{5}{4}x + 3 \]
So the left side is now:
\[ -\frac{5}{4}x + 3 \]
Step 2: Simplify the right side
Now, simplify the right side. First combine \(-\frac{1}{4}\) and \(5\):
\[ -\frac{1}{4} + 5 = -\frac{1}{4} + \frac{20}{4} = \frac{19}{4} \]
Now, combine this with \(\frac{1}{2}x\). Convert \(\frac{1}{2}x\) to have a common denominator with \(\frac{19}{4}\):
\[ \frac{1}{2}x = \frac{2}{4}x \]
So the right side becomes:
\[ \frac{2}{4}x + \frac{19}{4} = \frac{1}{2}x + \frac{19}{4} \]
Step 3: Write the simplified equation
Now we can rewrite the entire equation:
\[ -\frac{5}{4}x + 3 = \frac{1}{2}x + \frac{19}{4} \]
Step 4: Eliminate the fractions
To eliminate the fractions, multiply the whole equation by 4 (the least common denominator):
\[ 4 \left(-\frac{5}{4}x + 3\right) = 4 \left(\frac{1}{2}x + \frac{19}{4}\right) \]
This simplifies to:
\[ -5x + 12 = 2x + 19 \]
Step 5: Solve for \(x\)
Now, combine like terms:
- Add \(5x\) to both sides:
\[ 12 = 7x + 19 \]
- Subtract \(19\) from both sides:
\[ 12 - 19 = 7x \]
\[ -7 = 7x \]
- Divide both sides by \(7\):
\[ x = -1 \]
So the solution to the equation is:
\[ x = -1 \]