To determine which point satisfies the system of inequalities:
[tex]\[
\left\{\begin{array}{l}
y \leq 2x - 4 \\
y > -\frac{3}{4}x + 4
\end{array}\right.
\][/tex]
we need to check each point against both inequalities. Let's analyze each option in detail.
Point A. [tex]\((6, 0)\)[/tex]
1. Check the first inequality [tex]\( y \leq 2x - 4 \)[/tex]:
- Substitute [tex]\(x = 6\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[
0 \leq 2(6) - 4 \implies 0 \leq 12 - 4 \implies 0 \leq 8
\][/tex]
This inequality holds true.
2. Check the second inequality [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]:
- Substitute [tex]\(x = 6\)[/tex] and [tex]\(y = 0\)[/tex]:
[tex]\[
0 > -\frac{3}{4}(6) + 4 \implies 0 > -4.5 + 4 \implies 0 > -0.5
\][/tex]
This inequality also holds true.
Conclusion: Point [tex]\((6, 0)\)[/tex] satisfies both inequalities.
Point B. [tex]\((4, 1)\)[/tex]
1. Check the first inequality [tex]\( y \leq 2x - 4 \)[/tex]:
- Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
1 \leq 2(4) - 4 \implies 1 \leq 8 - 4 \implies 1 \leq 4
\][/tex]
This inequality holds true.
2. Check the second inequality [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]:
- Substitute [tex]\(x = 4\)[/tex] and [tex]\(y = 1\)[/tex]:
[tex]\[
1 > -\frac{3}{4}(4) + 4 \implies 1 > -3 + 4 \implies 1 > 1
\][/tex]
This inequality does not hold true.
Conclusion: Point [tex]\((4, 1)\)[/tex] does not satisfy both inequalities.
Point C. [tex]\((0, -4)\)[/tex]
1. Check the first inequality [tex]\( y \leq 2x - 4 \)[/tex]:
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[
-4 \leq 2(0) - 4 \implies -4 \leq -4
\][/tex]
This inequality holds true.
2. Check the second inequality [tex]\( y > -\frac{3}{4}x + 4 \)[/tex]:
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = -4\)[/tex]:
[tex]\[
-4 > -\frac{3}{4}(0) + 4 \implies -4 > 4
\][/tex]
This inequality does not hold true.
Conclusion: Point [tex]\((0, -4)\)[/tex] does not satisfy both inequalities.
Point D. [tex]\((0, 6)\)[/tex]
1. Check the first inequality [tex]\( y \leq 2x - 4 \)[/tex]:
- Substitute [tex]\(x = 0\)[/tex] and [tex]\(y = 6\)[/tex]:
[tex]\[
6 \leq 2(0) - 4 \implies 6 \leq -4
\][/tex]
This inequality does not hold true.
Conclusion: Point [tex]\((0, 6)\)[/tex] does not satisfy both inequalities.
Based on the analysis above, the point [tex]\( (6, 0) \)[/tex] satisfies both inequalities. Therefore, the answer is:
A. [tex]\( (6, 0) \)[/tex]
