Calculate Gabriela's Speed: Determining X for Her Last 6 km

To solve this problem, we'll follow these steps:

• Step 1: Calculate the time taken for each segment.
• Step 2: Use the average speed formula to connect these times and speed.
• Step 3: Solve the resulting equation for the unknown speed $v$.

Now, let's work through each step:

Step 1: Time for the first segment is $\frac{4}{8} = 0.5$ hours.

Time for the second segment is $\frac{6}{v}$ hours.

Step 2: Calculate the total time:

Total time $T = 0.5 + \frac{6}{v}$ hours.

The total distance is $4 + 6$ = 10 km.

The average speed given is $5x$ km/h, so:

$5x = \frac{10}{T} = \frac{10}{0.5 + \frac{6}{v}}$.

Solving for $v$:

Cross-multiply to clear the fraction:

$5x \left(0.5 + \frac{6}{v}\right) = 10$

Simplify:

$2.5x + \frac{30x}{v} = 10$

$\frac{30x}{v} = 10 - 2.5x$

$v = \frac{30x}{10 - 2.5x}$

Multiply numerator and denominator by 2 to simplify:

$v = \frac{60x}{20 - 5x}$

Further simplification:

$v = \frac{12x}{4 - x}$

Therefore, the solution to the problem is the speed during the last 6 km is $\frac{12x}{4-x}$ km/h.