Question

question 4
a number x does not exceed the sum of 30 and a number y. the difference of twice a number y and a number x is more than 12. the number x is at least 8. what system of inequalities models this situation?
\\(\

$$\begin{cases}x \\leq 30 + y \\\\ 2y - x > 12 \\\\ x \\geq 8\\end{cases}$$

\\)
\\(\

$$\begin{cases}x < 30 + y \\\\ 2y - x < 12 \\\\ x < 8\\end{cases}$$

\\)
\\(\

$$\begin{cases}x \\geq 30 + y \\\\ 2y - x \\geq 12 \\\\ x > 8\\end{cases}$$

\\)
\\(\

$$\begin{cases}x \\leq 30 + y \\\\ 2y - x \\geq 12 \\\\ x \\leq 8\\end{cases}$$

\\)

Explanation:

Step1: Translate first statement

"A number $x$ does not exceed the sum of 30 and $y$" means $x$ is less than or equal to $30+y$:
$x \leq 30 + y$

Step2: Translate second statement

"The difference of twice $y$ and $x$ is more than 12" means $2y - x$ is greater than 12:
$2y - x > 12$

Step3: Translate third statement

"$x$ is at least 8" means $x$ is greater than or equal to 8:
$x \geq 8$

Answer:

$\boldsymbol{x \leq 30 + y}$
$\boldsymbol{2y - x > 12}$
$\boldsymbol{x \geq 8}$
(This corresponds to the first option choice)