2. Quan plans to spend less than $80 on groceries. He intends to spend $68.25 on food and the remainder on juice. Each juice carton costs $3. He wants to know how many cartons of juice he can buy without exceeding his budget. (a) Let x represent the number of juice cartons Quan can purchase. Write an inequality to solve for x. (b) Solve the inequality and use it to determine how many juice cartons Quan can buy. Verify your answer.

(a) To write an inequality that represents the number of juice cartons Quan can purchase, we start with the total budget he plans not to exceed, which is $80. Quan plans to spend $68.25 on food, so we subtract that amount from the total budget to find the remaining amount he can spend on juice:

$80 - $68.25 = $11.75

Now, let x represent the number of juice cartons Quan can purchase. Since each carton costs $3, the total cost of x cartons of juice would be $3 times x, or $3x. Quan can spend up to $11.75 on juice, so this gives us the inequality:

$3x ≤ $11.75

(b) To solve the inequality for x, we divide both sides by $3:

$3x ≤ $11.75 x ≤ $11.75 / $3 x ≤ 3.9166...

Since Quan cannot buy a fraction of a carton, he can only buy whole cartons. Therefore, Quan can buy up to 3 whole cartons of juice without exceeding his budget.

Verification: If Quan buys 3 cartons of juice at $3 each, the total cost for juice would be:

3 cartons * $3/carton = $9

Adding this to the cost of food:

$68.25 (food) + $9 (juice) = $77.25

Since $77.25 is less than $80, Quan stays within his budget when he buys 3 cartons of juice.

Extra: This problem is an application of linear inequalities. An inequality describes the relationship between two expressions that may not be equal. In this case, we have an inequality that places an upper limit (budget) on what Quan can spend. When solving inequalities, it's very similar to solving equations. However, with inequalities, any multiplication or division by a negative number must be done carefully because it reverses the direction of the inequality (which was unnecessary in this problem since we only divided by a positive number).

The concept of rounding down to a whole number is important when dealing with items that cannot be divided, like juice cartons in this case. In mathematics, this is known as taking the "floor" of a number, which means rounding down to the nearest whole number.

In budgeting, it is essential to account for every item and stay within the total amount allocated. Using inequalities can be an effective way to manage one's budget and ensure that spending does not exceed the limit. It's especially useful because it gives a clear rule for what is allowable and what is not, just like the inequality produced in the example that demonstrates Quan's spending limit for the juice.