Jerry wants to purchase some items whose cost function is , where x is the number of items. If Jerry spends between $50 and $80, find the minimum and maximum number of items that he can purchase.

To find the minimum and maximum number of items Jerry can purchase within a spending range of $50 to $80, we need to analyze the cost function.

The cost function is given as C(x) = 3x + 10, where x represents the number of items Jerry wants to purchase.

To find the minimum number of items, we need to determine the lowest possible value for x that satisfies the spending range i.e., $50 ≤ C(x) ≤ $80.

Let's first find the minimum number of items Jerry can purchase:

$50 ≤ C(x) ≤ $80

Substituting C(x) = 3x + 10:

$50 ≤ 3x + 10 ≤ $80

To isolate the x variable, we subtract 10 from each part of the inequality:

$40 ≤ 3x ≤ $70

Next, divide through by 3 to solve for x:

$40/3 ≤ x ≤ $70/3

This implies:

13.333 ≤ x ≤ 23.333

Since we cannot purchase a fraction of an item, we need to round these values up or down to the nearest whole number:

Minimum number of items Jerry can purchase is 13.
(x = 13.333, round down to 13)

Now let's find the maximum number of items Jerry can purchase:

$50 ≤ C(x) ≤ $80

Substituting C(x) = 3x + 10:

$50 ≤ 3x + 10 ≤ $80

To isolate the x variable, we subtract 10 from each part of the inequality:

$40 ≤ 3x ≤ $70

Next, divide through by 3 to solve for x:

$40/3 ≤ x ≤ $70/3

This implies:

13.333 ≤ x ≤ 23.333

Since we cannot purchase a fraction of an item, we need to round these values up or down to the nearest whole number:

Maximum number of items Jerry can purchase is 23.
(x = 23.333, round up to 24)

Therefore, Jerry can purchase a minimum of 13 items and a maximum of 24 items within a spending range of $50 to $80.