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In the image, you are asked to solve for \( x \) such that lines \( A \) and \( B \) are parallel, denoted as \( A \parallel B \). The expressions for the angles at the intersection point between the transversal and lines \( A \) and \( B \) are given as:

- Line \( A \): \( 4x + 2 \)
- Line \( B \): \( 20x + 10 \)

For the lines \( A \) and \( B \) to be parallel, the corresponding angles must be equal. Therefore, we can set the two expressions equal to each other:

\[
4x + 2 = 20x + 10
\]

### Now, solving for \( x \):

1. Subtract \( 4x \) from both sides:

\[
2 = 16x + 10
\]

2. Subtract 10 from both sides:

\[
-8 = 16x
\]

3. Divide by 16:

\[
x = -\frac{1}{2}
\]

Thus, the value of \( x \) is \( -\frac{1}{2} \).

Let me know if you would like further details or explanations! Here are some follow-up questions:

1. What does it mean for two lines to be parallel in terms of their angles?
2. Can the value of \( x \) change if the lines were not parallel?
3. How would you verify that the lines are indeed parallel using geometry principles?
4. What other properties must be true for two lines to be considered parallel?
5. How would the equation change if you were solving for perpendicular lines instead of parallel lines?

**Tip**: When solving problems involving parallel lines and transversals, always look for corresponding, alternate interior, or alternate exterior angles, as these angles give important relationships between the lines.

Solve for x to make A parallel to B, where line A is represented by the expression 4x + 2 and line B is represented by the expression 20x + 10.

Learn how to solve for x to make lines A and B parallel. This problem involves algebraic manipulation and geometric concepts related to parallel lines and transversals. Suitable for students in Grades 7-9.

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