Given a segment from point 1 to point B with M at its midpoint,
AM - 2(3x - 1) AB - 5(2x + 4)
Find AB.

We are given a segment from point A (which corresponds to \( A(1) \)) to point B, with M as the midpoint of the segment \( AB \). We're also given two expressions:

- \( AM = 2(3x - 1) \)
- \( AB = 5(2x + 4) \)

We need to find the length of the segment \( AB \).

### Step 1: Relationship between \( AM \) and \( AB \)
Since M is the midpoint of \( AB \), the length of \( AM \) is half of \( AB \), meaning:

\[
AM = \frac{1}{2} AB
\]

### Step 2: Substitute the given expressions
Substitute the given expressions for \( AM \) and \( AB \) into the equation \( AM = \frac{1}{2} AB \):

\[
2(3x - 1) = \frac{1}{2} \times 5(2x + 4)
\]

### Step 3: Simplify both sides
- Left-hand side:

\[
2(3x - 1) = 6x - 2
\]

- Right-hand side:

\[
\frac{1}{2} \times 5(2x + 4) = \frac{5}{2} \times (2x + 4) = 5x + 10
\]

### Step 4: Set the simplified equation
Now, we have the equation:

\[
6x - 2 = 5x + 10
\]

### Step 5: Solve for \( x \)
To solve for \( x \), subtract \( 5x \) from both sides:

\[
6x - 5x - 2 = 10
\]

\[
x - 2 = 10
\]

Now, add 2 to both sides:

\[
x = 12
\]

### Step 6: Find \( AB \)
Now that we have \( x = 12 \), substitute this value into the expression for \( AB \):

\[
AB = 5(2x + 4) = 5(2(12) + 4) = 5(24 + 4) = 5 \times 28 = 140
\]

Thus, the length of the segment \( AB \) is \( 140 \).

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Here are some related questions to explore:
1. What is the formula for the midpoint of a segment in coordinate geometry?
2. How can you express the length of a

Given a segment from point A to point B with M at its midpoint, AM = 2(3x - 1), AB = 5(2x + 4). Find AB.

This problem demonstrates how to find the length of a segment AB given expressions for AM and AB. By applying the midpoint theorem and solving a linear equation, we find the length of AB step-by-step. Ideal for students in Grades 8-10, covering core concepts in algebra and geometry.

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