Let's start by solving the given equation:
\[ x + \frac{1}{2} = -\frac{1}{4} \]
To isolate \( x \), we can subtract \( \frac{1}{2} \) from both sides:
\[ x = -\frac{1}{4} - \frac{1}{2} \]
Now, we need to convert \( \frac{1}{2} \) to a fraction with a denominator of 4 for easier subtraction:
\[ \frac{1}{2} = \frac{2}{4} \]
So, we can rewrite the equation as:
\[ x = -\frac{1}{4} - \frac{2}{4} \]
Combining the fractions gives us:
\[ x = -\frac{3}{4} \]
Now we have found that the solution to the original equation is \( x = -\frac{3}{4} \).
Next, let's check which of the provided options has the same solution:
Option A: \[ x + \frac{1}{2} = \frac{5}{4} \] Subtract \( \frac{1}{2} \) (or \( \frac{2}{4} \)) from both sides: \[ x = \frac{5}{4} - \frac{2}{4} = \frac{3}{4} \] The solution is \( x = \frac{3}{4} \).
Option B: \[ x - 1.5 = 0.75 \] Add \( 1.5 \) to both sides. Remember that \( 1.5 \) can be written as \( \frac{3}{2} \) or \( \frac{6}{4} \): \[ x = 0.75 + 1.5 = 0.75 + \frac{6}{4} \] Convert \( 0.75 \) to a fraction: \[ 0.75 = \frac{3}{4} \] Now add the two fractions: \[ x = \frac{3}{4} + \frac{6}{4} = \frac{9}{4} \] The solution is \( x = \frac{9}{4} \).
Now, comparing the two results:
- Option A leads to \( x = \frac{3}{4} \).
- Option B leads to \( x = \frac{9}{4} \).
Neither option has the same solution as the original equation \( x + \frac{1}{2} = -\frac{1}{4} \), which solved to \( x = -\frac{3}{4} \). Thus, neither option is correct.