Question

6.10 systems of linear inequalities ws#1

1. determine if the following points are solutions to the system of inequalities.

a) (6, 0)
b) (4, -3)
challenge: c) (2, 1)
challenge: d) (3, 3)

1. determine if the following points are solutions to the system of inequalities.

a) (-4, 2)
b) (2, 5)
challenge: c) (4, 0)
challenge: d) (2, 1)
challenge: e) (0, -3)

Explanation:

First, we define the inequalities for each graph, then test each point:

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Problem 6 (Top Graph)

First, identify the inequalities from the graph:

1. Line 1: passes through (0, -3) and (3, 0), slope $m=1$, inequality: $y \geq x - 3$ (shaded above)
2. Line 2: passes through (0, 6) and (6, 0), slope $m=-1$, inequality: $y \geq -x + 6$ (shaded above)
3. Line 3: passes through (0, -6) and (6, -3), slope $m=\frac{1}{2}$, inequality: $y \leq \frac{1}{2}x - 6$ (shaded below)

The solution is the dark overlapping region, so a point must satisfy all three inequalities.

Step1: Test (6,0)

Substitute $x=6, y=0$:
$0 \geq 6-3=3$? No.

Step2: Test (4,-3)

Substitute $x=4, y=-3$:
$-3 \geq 4-3=1$? No.

Step3: Test (2,1)

Substitute $x=2, y=1$:
$1 \geq 2-3=-1$ ✔️; $1 \geq -2+6=4$? No.

Step4: Test (3,3)

Substitute $x=3, y=3$:
$3 \geq 3-3=0$ ✔️; $3 \geq -3+6=3$ ✔️; $3 \leq \frac{1}{2}(3)-6=-4.5$? No.

Step1: Test (-4,2)

Substitute $x=-4, y=2$:
$2 \geq 4+4=8$? No.

Step2: Test (2,5)

Substitute $x=2, y=5$:
$5 \geq -2+4=2$ ✔️; $5 \leq 2-4=-2$? No.

Step3: Test (4,0)

Substitute $x=4, y=0$:
$0 \geq -4+4=0$ ✔️; $0 \leq 4-4=0$ ✔️.

Step4: Test (2,1)

Substitute $x=2, y=1$:
$1 \geq -2+4=2$? No.

Step5: Test (0,-3)

Substitute $x=0, y=-3$:
$-3 \geq 0+4=4$? No.

Answer:

(Problem 6):
a) (6, 0): Not a solution
b) (4, -3): Not a solution
c) (2, 1): Not a solution
d) (3, 3): Not a solution

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Problem 7 (Bottom Graph)

Identify the inequalities from the graph:

1. Line 1: passes through (0,4) and (4,0), slope $m=-1$, inequality: $y \geq -x + 4$ (shaded above)
2. Line 2: passes through (0,-4) and (4,0), slope $m=1$, inequality: $y \leq x - 4$ (shaded below)

The solution is the dark overlapping region, so a point must satisfy both inequalities.