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?
$\

$$\begin{cases}a + b \\leq 30 \\\\ b \\geq a + 10\\end{cases}$$

$ $\

$$\begin{cases}a + b \\leq 30 \\\\ b \\leq a - 10\\end{cases}$$

$ $\

$$\begin{cases}a + b \\geq 30 \\\\ b \\geq a + 10\\end{cases}$$

$ $\

$$\begin{cases}a + b \\geq 30 \\\\ b \\leq a - 10\\end{cases}$$

$

Explanation:

Step1: Translate sum condition

The sum of $a$ and $b$ is at least 30, so:
$a + b \geq 30$

Step2: Translate difference condition

$b$ is greater, difference is at least 10:
$b - a \geq 10$, rearranged to $b \geq a + 10$

Step3: Match to options

Find the system with both inequalities.

Answer:

$\boldsymbol{a + b \geq 30}$
$\boldsymbol{b \geq a + 10}$