Let's first solve the given equation:
\[ x + \frac{1}{2} = -\frac{1}{4} a x + \frac{1}{2} \cdot \frac{5}{4} b x + 1.5 = 0.75 \]
However, it appears that the given equation has some formatting issues and seems to be a combination of equations. I'll assume you meant to say:
- \( x + \frac{1}{2} = -\frac{1}{4} ax \)
- \( \frac{5}{4} b x + 1.5 = 0.75 \)
Let's try to rearrange the first equation for clarity:
- \( x + \frac{1}{2} + \frac{1}{4} ax = 0 \)
Now, simplifying the second equation:
\[ \frac{5}{4} b x + 1.5 = 0.75 \] \[ \frac{5}{4} b x = 0.75 - 1.5 \] \[ \frac{5}{4} b x = -0.75 \]
Now, we rearranging them:
- For the first equation, let’s isolate \( x \):
\[ (1 + \frac{1}{4} a)x = -\frac{1}{2} \] \[ x = \frac{-\frac{1}{2}}{1 + \frac{1}{4} a} \]
- For the second equation:
\[ x = \frac{-0.75}{\frac{5}{4} b} \] \[ x = \frac{-0.75 \cdot 4}{5b} \] \[ x = \frac{-3}{5b} \]
The next step is to check each of the options (c and d) to see if any has the same solution.
Check Option C: \( x - \frac{1}{4} = 1 \)
Rearranging gives:
\[ x = 1 + \frac{1}{4} = \frac{5}{4} \]
Check Option D: \( x - 0.70 = -0.05 \)
Rearranging gives:
\[ x = 0.70 - 0.05 = 0.65 \]
Conclusion
Since we don't have variables \( a \) and \( b \) defined in those options and their full relation isn't clearly stated, we can't evaluate them directly against x. You need to either confirm the expression formats or clarify variable constraints.
However, from the equations you've checked, options C and D provide distinct solutions (5/4 and 0.65, respectively). Based on the algebra shown earlier, none can be confirmed without the actual values for a and b.
Please clarify the structure of the original equation if possible!