To determine which inequality pairs with \( y \leq -2x - 1 \) in the system of linear inequalities represented by the graph, we first analyze the two lines described:
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The first solid line with a negative slope goes through the points \((-2, 3)\) and \((0, -1)\). The equation given is \( y = -2x - 1 \), so the inequality associated with this line is \( y \leq -2x - 1 \) which means everything to the left of this line is shaded.
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The second dashed line goes through the points \((0, 2)\) and \((1, 0)\) which has a negative slope. To find its equation, we calculate the slope: \[ \text{slope} = \frac{0 - 2}{1 - 0} = -2 \] Therefore, the equation of the line can be expressed as \( y = -2x + 2 \).
Since the line is dashed, the inequality associated with this line does not include equality, which means it will be either \( y < -2x + 2 \) or \( y > -2x + 2 \).
Given that everything to the right of this line is shaded, the correct inequality must represent the region to the right of the line. Thus, it must be:
\[ y < -2x + 2 \]
So, the correct choice to complete the system of linear inequalities represented by the graph is:
\[ \boxed{y < -2x + 2} \]