Answer :
To determine which ordered pair satisfies both inequalities in the given system, we need to test each pair one by one against the inequalities:
[tex]\[ \begin{cases} y \geq \frac{2}{3}x + 1 \\ y < -\frac{1}{4}x + 2 \end{cases} \][/tex]
### Testing the Ordered Pair [tex]\((-6, 3.5)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 3.5 \geq \frac{2}{3}(-6) + 1 \][/tex]
[tex]\[ 3.5 \geq -4 + 1 \][/tex]
[tex]\[ 3.5 \geq -3 \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 3.5 < -\frac{1}{4}(-6) + 2 \][/tex]
[tex]\[ 3.5 < 1.5 + 2 \][/tex]
[tex]\[ 3.5 < 3.5 \][/tex]
This is false, as 3.5 is not less than 3.5.
Therefore, [tex]\((-6, 3.5)\)[/tex] does not satisfy both inequalities.
### Testing the Ordered Pair [tex]\((-6, -3)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ -3 \geq \frac{2}{3}(-6) + 1 \][/tex]
[tex]\[ -3 \geq -4 + 1 \][/tex]
[tex]\[ -3 \geq -3 \][/tex]
This is true, as [tex]\(-3 = -3\)[/tex].
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ -3 < -\frac{1}{4}(-6) + 2 \][/tex]
[tex]\[ -3 < 1.5 + 2 \][/tex]
[tex]\[ -3 < 3.5 \][/tex]
This is true.
Therefore, [tex]\((-6, -3)\)[/tex] satisfies both inequalities.
### Testing the Ordered Pair [tex]\((-4, 3)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 3 \geq \frac{2}{3}(-4) + 1 \][/tex]
[tex]\[ 3 \geq -\frac{8}{3} + 1 \][/tex]
[tex]\[ 3 \geq -\frac{8}{3} + \frac{3}{3} \][/tex]
[tex]\[ 3 \geq -\frac{5}{3} \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 3 < -\frac{1}{4}(-4) + 2 \][/tex]
[tex]\[ 3 < 1 + 2 \][/tex]
[tex]\[ 3 < 3 \][/tex]
This is false, as 3 is not less than 3.
Therefore, [tex]\((-4, 3)\)[/tex] does not satisfy both inequalities.
### Testing the Ordered Pair [tex]\((-4, 4)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 4 \geq \frac{2}{3}(-4) + 1 \][/tex]
[tex]\[ 4 \geq -\frac{8}{3} + 1 \][/tex]
[tex]\[ 4 \geq -\frac{8}{3} + \frac{3}{3} \][/tex]
[tex]\[ 4 \geq -\frac{5}{3} \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 4 < -\frac{1}{4}(-4) + 2 \][/tex]
[tex]\[ 4 < 1 + 2 \][/tex]
[tex]\[ 4 < 3 \][/tex]
This is false, as 4 is not less than 3.
Therefore, [tex]\((-4, 4)\)[/tex] does not satisfy both inequalities.
### Conclusion
The only ordered pair that satisfies both inequalities is [tex]\((-6, -3)\)[/tex].
[tex]\[ \begin{cases} y \geq \frac{2}{3}x + 1 \\ y < -\frac{1}{4}x + 2 \end{cases} \][/tex]
### Testing the Ordered Pair [tex]\((-6, 3.5)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 3.5 \geq \frac{2}{3}(-6) + 1 \][/tex]
[tex]\[ 3.5 \geq -4 + 1 \][/tex]
[tex]\[ 3.5 \geq -3 \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 3.5 < -\frac{1}{4}(-6) + 2 \][/tex]
[tex]\[ 3.5 < 1.5 + 2 \][/tex]
[tex]\[ 3.5 < 3.5 \][/tex]
This is false, as 3.5 is not less than 3.5.
Therefore, [tex]\((-6, 3.5)\)[/tex] does not satisfy both inequalities.
### Testing the Ordered Pair [tex]\((-6, -3)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ -3 \geq \frac{2}{3}(-6) + 1 \][/tex]
[tex]\[ -3 \geq -4 + 1 \][/tex]
[tex]\[ -3 \geq -3 \][/tex]
This is true, as [tex]\(-3 = -3\)[/tex].
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ -3 < -\frac{1}{4}(-6) + 2 \][/tex]
[tex]\[ -3 < 1.5 + 2 \][/tex]
[tex]\[ -3 < 3.5 \][/tex]
This is true.
Therefore, [tex]\((-6, -3)\)[/tex] satisfies both inequalities.
### Testing the Ordered Pair [tex]\((-4, 3)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 3 \geq \frac{2}{3}(-4) + 1 \][/tex]
[tex]\[ 3 \geq -\frac{8}{3} + 1 \][/tex]
[tex]\[ 3 \geq -\frac{8}{3} + \frac{3}{3} \][/tex]
[tex]\[ 3 \geq -\frac{5}{3} \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 3 < -\frac{1}{4}(-4) + 2 \][/tex]
[tex]\[ 3 < 1 + 2 \][/tex]
[tex]\[ 3 < 3 \][/tex]
This is false, as 3 is not less than 3.
Therefore, [tex]\((-4, 3)\)[/tex] does not satisfy both inequalities.
### Testing the Ordered Pair [tex]\((-4, 4)\)[/tex]
1. First Inequality: [tex]\( y \geq \frac{2}{3}x + 1 \)[/tex]
[tex]\[ 4 \geq \frac{2}{3}(-4) + 1 \][/tex]
[tex]\[ 4 \geq -\frac{8}{3} + 1 \][/tex]
[tex]\[ 4 \geq -\frac{8}{3} + \frac{3}{3} \][/tex]
[tex]\[ 4 \geq -\frac{5}{3} \][/tex]
This is true.
2. Second Inequality: [tex]\( y < -\frac{1}{4}x + 2 \)[/tex]
[tex]\[ 4 < -\frac{1}{4}(-4) + 2 \][/tex]
[tex]\[ 4 < 1 + 2 \][/tex]
[tex]\[ 4 < 3 \][/tex]
This is false, as 4 is not less than 3.
Therefore, [tex]\((-4, 4)\)[/tex] does not satisfy both inequalities.
### Conclusion
The only ordered pair that satisfies both inequalities is [tex]\((-6, -3)\)[/tex].
