Question

solve the systems of linear inequalities by graphing. graph the inequalities on your recording sheet. determine whether each ordered pair is a solution. \\(\

$$\begin{cases}2x - y \\geq 0 \\\\ 3x + y < 5\\end{cases}$$

\\) \\(\text{(-3,4)}\\) \\(\text{(0,0)}\\) \\(\text{(1,2)}\\) \\(\text{(1,-5)}\\) \\(\text{(5,8)}\\) \\(\text{(3,-8)}\\) \\(\text{(2,-7)}\\) \\(\text{(4,-7)}\\) \\(\text{solutions}\\) \\(\text{not solutions}\\)

Explanation:

Step1: Test (-3,4) in inequalities

First inequality: $2(-3) - 4 = -6 - 4 = -10 \geq 0$? No.
Second inequality: $3(-3) + 4 = -9 + 4 = -5 < 5$? Yes.
Fails first inequality, so not a solution.

Step2: Test (0,0) in inequalities

First inequality: $2(0) - 0 = 0 \geq 0$? Yes.
Second inequality: $3(0) + 0 = 0 < 5$? Yes.
Satisfies both, so solution.

Step3: Test (1,2) in inequalities

First inequality: $2(1) - 2 = 2 - 2 = 0 \geq 0$? Yes.
Second inequality: $3(1) + 2 = 3 + 2 = 5 < 5$? No.
Fails second inequality, so not a solution.

Step4: Test (1,-5) in inequalities

First inequality: $2(1) - (-5) = 2 + 5 = 7 \geq 0$? Yes.
Second inequality: $3(1) + (-5) = 3 - 5 = -2 < 5$? Yes.
Satisfies both, so solution.

Step5: Test (5,8) in inequalities

First inequality: $2(5) - 8 = 10 - 8 = 2 \geq 0$? Yes.
Second inequality: $3(5) + 8 = 15 + 8 = 23 < 5$? No.
Fails second inequality, so not a solution.

Step6: Test (3,-8) in inequalities

First inequality: $2(3) - (-8) = 6 + 8 = 14 \geq 0$? Yes.
Second inequality: $3(3) + (-8) = 9 - 8 = 1 < 5$? Yes.
Satisfies both, so solution.

Step7: Test (2,-7) in inequalities

First inequality: $2(2) - (-7) = 4 + 7 = 11 \geq 0$? Yes.
Second inequality: $3(2) + (-7) = 6 - 7 = -1 < 5$? Yes.
Satisfies both, so solution.

Step8: Test (4,-7) in inequalities

First inequality: $2(4) - (-7) = 8 + 7 = 15 \geq 0$? Yes.
Second inequality: $3(4) + (-7) = 12 - 7 = 5 < 5$? No.
Fails second inequality, so not a solution.

Answer:

SOLUTIONS:

$(0, 0)$, $(1, -5)$, $(3, -8)$, $(2, -7)$

NOT SOLUTIONS:

$(-3, 4)$, $(1, 2)$, $(5, 8)$, $(4, -7)$