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]
