If the distance of 𝐴((2𝑥 − 3), 5) from line 𝑥 = −4 is equal to 7, then find the value of x

We are given a point \( A(2x - 3, 5) \) and the line \( x = -4 \). The problem states that the distance from point \( A \) to the line is equal to 7. We need to find the value of \( x \).

### Step-by-Step Solution:

1. The equation of the line is \( x = -4 \), which is a vertical line. The distance from any point to a vertical line depends only on the x-coordinates of the point and the line.
   
2. The formula to calculate the distance from a point \( (x_1, y_1) \) to a vertical line \( x = c \) is:
   \[
   d = |x_1 - c|
   \]
   where \( x_1 \) is the x-coordinate of the point and \( c \) is the x-value of the line.

3. For the given point \( A(2x - 3, 5) \), the x-coordinate is \( 2x - 3 \), and the line is \( x = -4 \). Substituting into the distance formula:
   \[
   d = |(2x - 3) - (-4)| = |2x - 3 + 4| = |2x + 1|
   \]

4. According to the problem, the distance is given as 7. So we set up the equation:
   \[
   |2x + 1| = 7
   \]

5. Now, solve for \( x \) by splitting into two cases:
   
   **Case 1:**
   \[
   2x + 1 = 7
   \]
   \[
   2x = 6 \quad \Rightarrow \quad x = 3
   \]

   **Case 2:**
   \[
   2x + 1 = -7
   \]
   \[
   2x = -8 \quad \Rightarrow \quad x = -4
   \]

### Final Answer:
The possible values of \( x \) are \( 3 \) and \( -4 \).

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How do you calculate the distance between two points on a coordinate plane?
2. What is the general formula for the distance from a point to a line in 2D geometry?
3. Can you explain how absolute value equations are solved in general?
4. How would the solution change if the given line was horizontal instead of vertical?
5. What is the significance of a point being equidistant from a line in geometric constructions?

**Tip**: When dealing with absolute value equations, always consider both the positive and negative cases.

If the distance of A((2x − 3), 5) from line x = −4 is equal to 7, then find the value of x.

Solve for the value of x when the distance from point A(2x − 3, 5) to the line x = −4 is 7. Explore concepts like distance formulas, absolute value equations, and linear equations. Ideal for Grades 8-10.

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