The sum of two positive integers, [tex]\( a \)[/tex] and [tex]\( b \)[/tex], is at least 30. The difference of the two integers is at least 10. If [tex]\( b \)[/tex] is the greater integer, which system of inequalities could represent the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex]?

A. [tex]\( a + b \geq 30 \quad b \geq a + 10 \)[/tex]

B. [tex]\( a + b \geq 30 \quad b \leq a - 10 \)[/tex]

C. [tex]\( a + b \leq 30 \quad b \geq a + 10 \)[/tex]

D. [tex]\( a + b \leq 30 \quad b \leq a - 10 \)[/tex]



Answer :

To determine which system of inequalities correctly represents the given conditions, let's carefully analyze each condition step-by-step:

1. Condition 1: The sum of two positive integers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] is at least 30.
This translates to the inequality:
[tex]\[ a + b \geq 30 \][/tex]

2. Condition 2: The difference of the two integers is at least 10.
Since [tex]\(b\)[/tex] is the greater integer, this translates to:
[tex]\[ b \geq a + 10 \][/tex]

Now let's compare these inequalities with the given options:

- Option 1: [tex]\(a + b \geq 30 \quad b \geq a + 10\)[/tex]
- This option satisfies both conditions:
- The sum condition: [tex]\(a + b \geq 30\)[/tex]
- The difference condition: [tex]\(b \geq a + 10\)[/tex]

- Option 2: [tex]\(a + b \geq 30 \quad b \leq a - 10\)[/tex]
- This option does not make sense because it contradicts the condition that [tex]\(b\)[/tex] is the greater integer. If [tex]\(b \leq a - 10\)[/tex], then [tex]\(b\)[/tex] cannot be greater than [tex]\(a\)[/tex].

- Option 3: [tex]\(a + b \leq 30 \quad b \geq a + 10\)[/tex]
- This option contradicts the sum condition, as the sum is supposed to be at least 30. This inequality [tex]\(a + b \leq 30\)[/tex] is invalid.

- Option 4: [tex]\(a + b \leq 30 \quad b \leq a - 10\)[/tex]
- Similar to Option 3, this option contradicts the sum condition. Furthermore, it also contradicts the condition that [tex]\(b\)[/tex] is supposed to be greater than [tex]\(a\)[/tex].

Hence, the correct system of inequalities that represents the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] based on the provided conditions is:
[tex]\[ a + b \geq 30 \quad b \geq a + 10 \][/tex]