To determine how many cases of soda Philip can buy while remaining under his budget, we start with the amount he has already spent and subtract that from his budget to find out how much he has left to spend.
Philip's budget is $496, and he has already spent $400. Thus, the remaining amount he can spend is:
\[ 496 - 400 = 96 \]
Next, let \( x \) represent the number of cases of soda he can purchase. Each case costs $8, so the total cost for \( x \) cases is \( 8x \). To stay within his remaining budget of $96, we need to set up the following inequality:
\[ 8x < 96 \]
Now, we will solve for \( x \):
\[ x < \frac{96}{8} \] \[ x < 12 \]
Since \( x \) must be a whole number (you can't purchase a fraction of a case), the maximum value for \( x \) is 11. Therefore, Philip can buy up to 11 cases of soda to stay under his budget of $496.
Number line representation:
In the number line that represents the cases of soda Philip can purchase:
- The values would range from 0 to 11, indicating the maximum number of cases he can buy is 11.
- We would mark values 0 through 11 and indicate that 11 is the largest whole number he can buy while staying under his budget.
Thus, the largest integer \( x \) satisfying the budget constraint is:
The number of cases of soda Philip can buy is \( x \leq 11 \).
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