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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Do you want any further details or have any questions?

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