Question

the sum of two positive integers, a and b, is at least 30. the difference of the two integers is at least 10. if b is the greater integer, which system of inequalities could represent the values of a and b?
a + b ≥ 30 b ≥ a + 10
a + b ≥ 30 b ≤ a − 10
a + b ≤ 30 b ≥ a + 10
a + b ≤ 30 b ≤ a − 10

Explanation:

Step1: Analyze the sum condition

The sum of two positive integers \(a\) and \(b\) is at least 30. "At least" means greater than or equal to, so the inequality for the sum is \(a + b\geq30\).

Step2: Analyze the difference condition

The difference of the two integers (with \(b\) being the greater integer) is at least 10. The difference \(b - a\) should be at least 10, which can be written as \(b - a\geq10\), and rearranging this gives \(b\geq a + 10\).

Answer:

A. \(a + b\geq30\) \(b\geq a + 10\)