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Graph the system of inequalities and determine which points are solutions.

[tex]\[
\begin{array}{l}
y \leq -x + 1 \\
y \ \textgreater \ x
\end{array}
\][/tex]

Check the following points:

1. [tex]\((-3, 5)\)[/tex]
2. [tex]\((-2, 2)\)[/tex]
3. [tex]\((-1, -3)\)[/tex]


Sagot :

Let's analyze each point [tex]\((-3, 5)\)[/tex], [tex]\((-2, 2)\)[/tex], and [tex]\((-1, -3)\)[/tex] against the given inequalities:

1. First Inequality: [tex]\( y \leq -x + 1 \)[/tex]
2. Second Inequality: [tex]\( y > x \)[/tex]

### Point: [tex]\((-3, 5)\)[/tex]

- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]

Substitute [tex]\( x = -3 \)[/tex] and [tex]\( y = 5 \)[/tex]:

[tex]\[ 5 \leq -(-3) + 1 \implies 5 \leq 3 + 1 \implies 5 \leq 4 \quad \text{(False)} \][/tex]

Since this point does not satisfy the first inequality, we do not need to check the second one.

### Point: [tex]\((-2, 2)\)[/tex]

- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]

Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 2 \)[/tex]:

[tex]\[ 2 \leq -(-2) + 1 \implies 2 \leq 2 + 1 \implies 2 \leq 3 \quad \text{(True)} \][/tex]

The point satisfies the first inequality.

- Second Inequality: [tex]\( y > x \)[/tex]

Substitute [tex]\( x = -2 \)[/tex] and [tex]\( y = 2 \)[/tex]:

[tex]\[ 2 > -2 \quad \text{(True)} \][/tex]

The point also satisfies the second inequality.

Since this point satisfies both inequalities, we record [tex]\(1\)[/tex] for this point.

### Point: [tex]\((-1, -3)\)[/tex]

- First Inequality: [tex]\( y \leq -x + 1 \)[/tex]

Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -3 \)[/tex]:

[tex]\[ -3 \leq -(-1) + 1 \implies -3 \leq 1 + 1 \implies -3 \leq 2 \quad \text{(True)} \][/tex]

The point satisfies the first inequality.

- Second Inequality: [tex]\( y > x \)[/tex]

Substitute [tex]\( x = -1 \)[/tex] and [tex]\( y = -3 \)[/tex]:

[tex]\[ -3 > -1 \quad \text{(False)} \][/tex]

Since this point does not satisfy the second inequality, we record [tex]\(0\)[/tex] for this point.

### Summary

- For the point [tex]\((-3, 5)\)[/tex], neither inequality is satisfied → 0
- For the point [tex]\((-2, 2)\)[/tex], both inequalities are satisfied → 1
- For the point [tex]\((-1, -3)\)[/tex], only one inequality is satisfied → 0

Therefore, the final result for the points [tex]\((-3, 5)\)[/tex], [tex]\((-2, 2)\)[/tex], and [tex]\((-1, -3)\)[/tex] is:

[tex]\[ [0, 1, 0] \][/tex]
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