Let's solve the equation step by step. The given equation is:
\[ \frac{3}{4}x + 3 - 2x = -\frac{1}{4} + \frac{1}{2}x + 5 \]
Step 1: Combine like terms
Let's first combine the constant terms and the x terms on both sides.
Left side:
\(\frac{3}{4}x - 2x\) can be written with a common denominator (4):
\[ \frac{3}{4}x - \frac{8}{4}x = -\frac{5}{4}x \]
So the left side becomes:
\[ -\frac{5}{4}x + 3 \]
Right side:
Combine the constants \(-\frac{1}{4} + 5\):
To combine these, you can convert 5 into a fraction with a denominator of 4:
\[ 5 = \frac{20}{4} \]
So,
\[ -\frac{1}{4} + \frac{20}{4} = \frac{19}{4} \]
Now, you have:
\[ \frac{1}{2}x = \frac{1}{2}x \]
The equation now looks like this:
\[ -\frac{5}{4}x + 3 = \frac{1}{2}x + \frac{19}{4} \]
Step 2: Isolate the variable
Now, let's isolate the x terms on one side. We can add \(\frac{5}{4}x\) to both sides and subtract \(\frac{19}{4}\) from both sides:
\[ 3 - \frac{19}{4} = \frac{1}{2}x + \frac{5}{4}x \]
Converting 3 into a fraction with a denominator of 4:
\[ 3 = \frac{12}{4} \]
So,
\[ \frac{12}{4} - \frac{19}{4} = \frac{1}{2}x + \frac{5}{4}x \]
Step 3: Simplifying both sides
The left side simplifies to:
\[ \frac{12 - 19}{4} = -\frac{7}{4} \]
On the right side, combine the x terms:
Convert \(\frac{1}{2}x\) into a fraction with a denominator of 4:
\[ \frac{1}{2}x = \frac{2}{4}x \]
So,
\[ \frac{2}{4}x + \frac{5}{4}x = \frac{7}{4}x \]
Step 4: Final equation
Now you have:
\[ -\frac{7}{4} = \frac{7}{4}x \]
To solve for \(x\), multiply both sides by \(\frac{4}{7}\):
\[ x = -1 \]
And that is the solution!