To match the inequalities to their graphs, let's break down the given inequalities:
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Inequality: \( y \geq x + 1 \)
- This graph includes the line \( y = x + 1 \) and the area above it (including the line).
- Is (0,0) a solution? No, because \( 0 \) is not greater than or equal to \( 0 + 1 \).
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Inequality: \( y < x + 1 \)
- This graph includes the area below the line \( y = x + 1 \) (but not the line itself).
- Is (0,0) a solution? Yes, because \( 0 < 0 + 1 \).
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Inequality: \( y > x + 1 \)
- This graph includes the area above the line \( y = x + 1 \) (but not the line itself).
- Is (0,0) a solution? No, because \( 0 \) is not greater than \( 0 + 1 \).
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Inequality: \( y \leq x + 1 \)
- This graph includes the line \( y = x + 1 \) and the area below it (including the line).
- Is (0,0) a solution? Yes, because \( 0 \leq 0 + 1 \).
Summary:
- For \( y \geq x + 1 \): No, (0,0) is not a solution.
- For \( y < x + 1 \): Yes, (0,0) is a solution.
- For \( y > x + 1 \): No, (0,0) is not a solution.
- For \( y \leq x + 1 \): Yes, (0,0) is a solution.