Question

writing an inequality in two variables which linear inequality could represent the given table of values? \\(\circ\\) \\(y < -2x + 3\\) \\(\circ\\) \\(y \leq -2x + 3\\) \\(\circ\\) \\(y > -\frac{1}{2}x - 3\\) \\(\circ\\) \\(y \leq -\frac{1}{2}x - 3\\) \\(\

$$\begin{array}{|c|c|} \\hline x & y \\\\ \\hline -4 & -1 \\\\ \\hline -2 & 4 \\\\ \\hline 3 & -3 \\\\ \\hline 3 & -4 \\\\ \\hline \\end{array}$$

\\)

Explanation:

Step1: Test first point in options

Test $x=-4, y=-1$ in each inequality:

1. $-1 < -2(-4)+3 \implies -1 < 11$ (True)
2. $-1 \leq -2(-4)+3 \implies -1 \leq 11$ (True)
3. $-1 > -\frac{1}{2}(-4)-3 \implies -1 > -1$ (False)
4. $-1 \leq -\frac{1}{2}(-4)-3 \implies -1 \leq -1$ (True)

Eliminate option 3.

Step2: Test second point in options

Test $x=-2, y=4$ in remaining inequalities:

1. $4 < -2(-2)+3 \implies 4 < 7$ (True)
2. $4 \leq -2(-2)+3 \implies 4 \leq 7$ (True)
3. $4 \leq -\frac{1}{2}(-2)-3 \implies 4 \leq -2$ (False)

Eliminate option 4.

Step3: Test third point in options

Test $x=3, y=-3$ in remaining inequalities:

1. $-3 < -2(3)+3 \implies -3 < -3$ (False)
2. $-3 \leq -2(3)+3 \implies -3 \leq -3$ (True)

Eliminate option 1.

Step4: Verify fourth point

Test $x=3, y=-4$ in $y \leq -2x+3$:
$-4 \leq -2(3)+3 \implies -4 \leq -3$ (True)

Answer:

$\boldsymbol{y \leq -2x + 3}$