Solving the Absolute Value Equation $|4X - 7| = 5$

The problem asks for the values of the expression $2 |X| – | – X|$ given the equation $|4X - 7| = 5$. First, we need to solve the absolute value equation for $X$.

1. Split the equation: An absolute value equation $|A| = B$ implies $A = B$ or $A = -B$. So, $|4X - 7| = 5$ means:

   • Case 1: $4X - 7 = 5$
   • Case 2: $4X - 7 = -5$

2. Solve Case 1:

   $4X - 7 = 5$

   Add 7 to both sides:

   $4X = 5 + 7$

   $4X = 12$

   Divide by 4:

   $X = \frac{12}{4}$

   $X = 3$

3. Solve Case 2:

   $4X - 7 = -5$

   Add 7 to both sides:

   $4X = -5 + 7$

   $4X = 2$

   Divide by 4:

   $X = \frac{2}{4}$

   $X = \frac{1}{2}$

   So, the possible values for $X$ are $3$ and $\frac{1}{2}$.

4. Simplify the expression: The expression to evaluate is $2 |X| – | – X|$. Recall the property of absolute values: $|-a| = |a|$. Applying this, we get:

   $2 |X| – | – X| = 2 |X| - |X| = |X|$

   The expression simplifies to just $|X|$.

5. Evaluate the simplified expression for each value of X:

   • If $X = 3$, then $|X| = |3| = 3$.
   • If $X = \frac{1}{2}$, then $|X| = |\frac{1}{2}| = \frac{1}{2}$.

   Therefore, the possible values for the expression $2 |X| – | – X|$ are $3$ and $\frac{1}{2}$.

Comparing this result with the options, Option B contains the values $\frac{1}{2}$ and $3$.