Question

which of the following are solutions to the system of inequalities below? select all that are solutions.\

$$\begin{cases} y \\leq 2x + 3 \\\\ 5x + 2y > 15 \\end{cases}$$

\
\\(\square\\) \\((-2, 2)\\)\
\\(\square\\) \\((2,0)\\)\
\\(\square\\) \\((1,5)\\)\
\\(\square\\) \\((3,9)\\)\
\\(\square\\) \\((4,4)\\)\
\\(\square\\) \\((3,0)\\)

Explanation:

Step1: Test (-2,2) in inequalities

First inequality: $2 \leq 2(-2)+3 \implies 2 \leq -1$ (False)
Second inequality: $5(-2)+2(2) > 15 \implies -10+4 >15 \implies -6>15$ (False)

Step2: Test (2,0) in inequalities

First inequality: $0 \leq 2(2)+3 \implies 0 \leq 7$ (True)
Second inequality: $5(2)+2(0) >15 \implies 10>15$ (False)

Step3: Test (1,5) in inequalities

First inequality: $5 \leq 2(1)+3 \implies 5 \leq 5$ (True)
Second inequality: $5(1)+2(5) >15 \implies 5+10>15 \implies 15>15$ (False)

Step4: Test (3,9) in inequalities

First inequality: $9 \leq 2(3)+3 \implies 9 \leq 9$ (True)
Second inequality: $5(3)+2(9) >15 \implies 15+18>15 \implies 33>15$ (True)

Step5: Test (4,4) in inequalities

First inequality: $4 \leq 2(4)+3 \implies 4 \leq 11$ (True)
Second inequality: $5(4)+2(4) >15 \implies 20+8>15 \implies 28>15$ (True)

Step6: Test (3,0) in inequalities

First inequality: $0 \leq 2(3)+3 \implies 0 \leq 9$ (True)
Second inequality: $5(3)+2(0) >15 \implies 15>15$ (False)

Answer:

(3,9), (4,4)