The inequality \( y \leq -2x - 1 \) represents a linear inequality in two variables. To solve it, we need to graph the corresponding line and shade the region that satisfies the inequality.
Graph the line \( y = -2x - 1 \).
The line \( y = -2x - 1 \) has a slope of \(-2\) and a y-intercept at \((0, -1)\). Plot the y-intercept and use the slope to find another point on the line. For example, when \( x = 1 \), \( y = -2(1) - 1 = -3 \). So, the line passes through \((0, -1)\) and \((1, -3)\).
Determine the region to shade.
Since the inequality is \( y \leq -2x - 1 \), we shade the region below the line \( y = -2x - 1 \). This includes all points on the line itself because the inequality is "less than or equal to."
The solution to the inequality \( y \leq -2x - 1 \) is the region below and including the line \( y = -2x - 1 \). \\(\boxed{\text{Shade the region below the line } y = -2x - 1}\\)
The solution to the inequality \( y \leq -2x - 1 \) is the region below and including the line \( y = -2x - 1 \). \\(\boxed{\text{Shade the region below the line } y = -2x - 1}\\)
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