Absolute Value Equation Problem Analysis

The problem asks us to first solve an absolute value equation to find the possible values of X, and then substitute these values into a given expression to find its corresponding numerical results. The core concepts involved are understanding absolute value and simplifying algebraic expressions.

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

An absolute value equation of the form $|A| = B$ means that $A$ can be either $B$ or $-B$. In this problem, $A = 4X - 7$ and $B = 5$. Therefore, we will have two separate cases to solve for X:

• Case 1: The expression inside the absolute value is equal to the positive value.
• Case 2: The expression inside the absolute value is equal to the negative value.

Case 1: $4X - 7 = 5$

To solve for X, we will isolate X on one side of the equation:

• Add 7 to both sides:
• $4X - 7 + 7 = 5 + 7$
• $4X = 12$
• Divide both sides by 4:
• $X = \frac{12}{4}$
• $X = 3$

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

Similarly, we isolate X for this case:

• Add 7 to both sides:
• $4X - 7 + 7 = -5 + 7$
• $4X = 2$
• Divide both sides by 4:
• $X = \frac{2}{4}$
• $X = \frac{1}{2}$

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

Evaluating the Expression $2 |X| - | - X|$

Now that we have the values of X, we need to calculate the value of the expression $2 |X| - | - X|$ for each of these X values.

First, let's simplify the expression. Remember a key property of absolute values: $|-a| = |a|$. Using this property, we can simplify $|-X|$ to $|X|$.

The expression becomes:

$2 |X| - |X|$

By combining like terms, we get:

$|X|$

So, for each value of X, the value of the expression $2 |X| - | - X|$ is simply $|X|$.

Calculating for $X = 3$

When $X = 3$, we substitute this value into the simplified expression:

$|X| = |3| = 3$

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

When $X = \frac{1}{2}$, we substitute this value into the simplified expression:

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

Final Values of the Expression

Based on our calculations, the values of $2 |X| - | - X|$ corresponding to the two possible values of X are $3$ and $\frac{1}{2}$.

The solution values are $\frac{1}{2}$ and $3$.

We can summarize the results in a table: