To solve this problem, we need to determine the value of x, which represents the increased allowance Francie receives per week.
Let's break down the information given:
Initial allowance per week: $5
Number of weeks for the initial allowance: 8
Increased allowance per week: $x
Number of weeks for the increased allowance: 6
Amount of money spent on new clothes: 1/2 * (Total allowance)
Cost of the video game: $35
Amount of money remaining after buying the video game: $3
We can set up the equation as follows:
(Initial allowance per week * Number of weeks for the initial allowance) + (Increased allowance per week * Number of weeks for the increased allowance) - (1/2 * (Total allowance)) - Cost of the video game = Amount of money remaining
($5 * 8) + ($x * 6) - (1/2 * (($5 * 8) + ($x * 6))) - $35 = $3
Let's simplify and solve for x:
40 + 6x - (1/2 * (40 + 6x)) - 35 = 3
40 + 6x - (20 + 3x) - 35 = 3
40 + 6x - 20 - 3x - 35 = 3
3x - 15 = 3
To isolate x, we add 15 to both sides of the equation:
3x - 15 + 15 = 3 + 15
3x = 18
To solve for x, we divide both sides of the equation by 3:
x = 18 / 3
x = 6
The value of x is 6.
#### 6
The answer is: 6
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
Francie saves up her allowance for several weeks. She receives an allowance of $5 a week for 8 weeks. Then her dad raises her allowance, and she receives $x a week for 6 weeks. Francie uses half of the money to buy new clothes. With the remaining money, she buys a video game that costs $35. Francie has 3 money remaining after buying the video game.

### Answer:
