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Which points are solutions to the system of inequalities shown below?

[tex]\[
\begin{array}{l}
y \ \textless \ x \\
y \ \textless \ 5 \\
x \ \textless \ 4
\end{array}
\][/tex]

Check all that apply.

A. [tex]\((3,0)\)[/tex]
B. [tex]\((3,4)\)[/tex]
C. [tex]\((6,2)\)[/tex]
D. [tex]\((2,1)\)[/tex]
E. [tex]\((0,-1)\)[/tex]
F. [tex]\((1,7)\)[/tex]

Sagot :

GinnyAnswer
To determine which points satisfy the given system of inequalities, we need to check each point against the three inequalities:

[tex]\[ \begin{array}{l} y < x \\ y < 5 \\ x < 4 \end{array} \][/tex]

Let's check each given point one-by-one:

Option A: [tex]\((3, 0)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(0 < 3\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(0 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(3 < 4\)[/tex] (True)

Since all three inequalities are satisfied, [tex]\((3, 0)\)[/tex] is a solution.

Option B: [tex]\((3, 4)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(4 < 3\)[/tex] (False)
2. [tex]\(y < 5\)[/tex]: [tex]\(4 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(3 < 4\)[/tex] (True)

Since the first inequality is not satisfied, [tex]\((3, 4)\)[/tex] is not a solution.

Option C: [tex]\((6, 2)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(2 < 6\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(2 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(6 < 4\)[/tex] (False)

Since the third inequality is not satisfied, [tex]\((6, 2)\)[/tex] is not a solution.

Option D: [tex]\((2, 1)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(1 < 2\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(1 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(2 < 4\)[/tex] (True)

Since all three inequalities are satisfied, [tex]\((2, 1)\)[/tex] is a solution.

Option E: [tex]\((0, -1)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(-1 < 0\)[/tex] (True)
2. [tex]\(y < 5\)[/tex]: [tex]\(-1 < 5\)[/tex] (True)
3. [tex]\(x < 4\)[/tex]: [tex]\(0 < 4\)[/tex] (True)

Since all three inequalities are satisfied, [tex]\((0, -1)\)[/tex] is a solution.

Option F: [tex]\((1, 7)\)[/tex]

1. [tex]\(y < x\)[/tex]: [tex]\(7 < 1\)[/tex] (False)
2. [tex]\(y < 5\)[/tex]: [tex]\(7 < 5\)[/tex] (False)
3. [tex]\(x < 4\)[/tex]: [tex]\(1 < 4\)[/tex] (True)

Since the first and second inequalities are not satisfied, [tex]\((1, 7)\)[/tex] is not a solution.

After checking all points, the solutions to the system of inequalities are:

A. [tex]\((3, 0)\)[/tex], D. [tex]\((2, 1)\)[/tex], E. [tex]\((0, -1)\)[/tex]
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