In $\Delta DEF$, AB $\parallel$ EF. The value of x is :
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In geometry problems, always double-check your solutions to ensure they don't produce zero or negative lengths.
If you solve the equation as a full quadratic:
$2x(2x - 0.5) = x(3x + 1) \implies 4x^2 - x = 3x^2 + x \implies x^2 - 2x = 0 \implies x(x-2) = 0$.
This gives two mathematical roots, $x = 0$ and $x = 2$.
However, $x = 0$ is rejected because side lengths in geometry must be strictly positive.
Step 1: Write down the proportion from the Basic Proportionality Theorem.
Since $AB \parallel EF$ in $\Delta DEF$, by the Basic Proportionality Theorem:
\[ \frac{DA}{AE} = \frac{DB}{BF} \]
From the figure, $DA = 2x$, $AE = 3x+1$, $DB = x$, and $BF = 2x - \frac{1}{2}$.
Step 2: Cross multiply the full expressions right away, without dividing by x first.
\[ \frac{2x}{3x+1} = \frac{x}{2x - \frac{1}{2}} \]
Cross multiplying:
\[ 2x\left(2x - \frac{1}{2}\right) = x(3x+1) \]
Step 3: Expand both sides completely.
\[ 4x^2 - x = 3x^2 + x \]
Step 4: Bring every term to one side and simplify into a quadratic in x.
\[ 4x^2 - x - 3x^2 - x = 0 \]
\[ x^2 - 2x = 0 \]
Step 5: Factor out the common term x.
\[ x(x - 2) = 0 \]
This gives two possible solutions:
\[ x = 0 \quad \text{or} \quad x = 2 \]
Step 6: Reject the value that does not make geometric sense.
If $x = 0$, then $DA = 2(0) = 0$ and $DB = 0$, meaning points $A$ and $B$ would coincide with $D$, which is not possible for a line segment $AB$ drawn inside the triangle. So $x = 0$ must be rejected, leaving only $x = 2$.
Final Answer:
The only valid value is $x = 2$, which matches Option (B).
\[ \boxed{x = 2} \]
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