Составление линейных неравенств в стандартной форме

Task 1

Here's how to solve the problem:

The problem asks us to write an inequality in standard form to represent the given situation.

1. Identify the variables:
* x = the number of standard containers
* y = the number of large containers

2. Identify the costs:
* Cost of a standard container = $1,300
* Cost of a large container = $3,100

3. Identify the total budget constraint:
* The total expense must be under $53,000. This means the total cost must be less than $53,000.

4. Formulate the expression for the total cost:
The total cost is the sum of the cost of standard containers and the cost of large containers.
Total Cost = (Cost per standard container * number of standard containers) + (Cost per large container * number of large containers)
Total Cost = \(1300x + 3100y\)

5. Write the inequality:
Since the total expense must be under $53,000, we can write the inequality as:
\(1300x + 3100y < 53000\)

6. Standard Form:
The problem asks for the inequality in standard form. For linear inequalities in two variables, the standard form is typically \(Ax + By < C\) or \(Ax + By \le C\) (or with greater than symbols), where A, B, and C are integers, and A is usually non-negative.

Our inequality \(1300x + 3100y < 53000\) is already in this form. However, we can simplify it by dividing all terms by a common factor. The greatest common divisor of 1300, 3100, and 53000 is 100.

Divide each term by 100:
\(\frac{1300x}{100} + \frac{3100y}{100} < \frac{53000}{100}\)
\(13x + 31y < 530\)

Final Answer:
The inequality in standard form is \(13x + 31y < 530\).