Question

write the following sentences as a system of inequalities in two variables. then graph the system. the sum of the x-variable and the y-variable is at most 6. the y-variable added to the product of 2 and the x-variable does not exceed 6. choose the correct system of inequalities. a. $x + y \geq 6$ $2x + y \leq 6$ b. $x + y \geq 6$ $2x + y \geq 6$ c. $x + y \leq 6$ $2x + y \geq 6$ d. $x + y \leq 6$ $2x + y \leq 6$

Explanation:

Step1: Analyze the first sentence

The sum of \( x \)-variable and \( y \)-variable is at most 6. "At most" means less than or equal to, so the inequality is \( x + y \leq 6 \).

Step2: Analyze the second sentence

The \( y \)-variable added to the product of 2 and \( x \)-variable does not exceed 6. "Does not exceed" means less than or equal to, so the inequality is \( 2x + y \leq 6 \).

Step3: Match with options

Looking at the options, option D has \( x + y \leq 6 \) and \( 2x + y \leq 6 \), which matches our derived inequalities.

Answer:

D. \( x + y \leq 6 \)
\( 2x + y \leq 6 \)