1. **State the problem:** We need to graph the union of the inequalities $x + y \leq -5$ or $x - y \geq 7$. 2. **Understand the inequalities:** - The first inequality $x + y \leq -5$ represents all points $(x,y)$ below or on the line $x + y = -5$. - The second inequality $x - y \geq 7$ represents all points $(x,y)$ above or on the line $x - y = 7$. 3. **Rewrite each line in slope-intercept form:** - For $x + y = -5$, solve for $y$: $$y = -x - 5$$ - For $x - y = 7$, solve for $y$: $$-y = 7 - x \implies y = x - 7$$ 4. **Graph the lines:** - The line $y = -x - 5$ has slope $-1$ and y-intercept $-5$. - The line $y = x - 7$ has slope $1$ and y-intercept $-7$. 5. **Shade the regions:** - For $x + y \leq -5$, shade below or on the line $y = -x - 5$. - For $x - y \geq 7$, shade above or on the line $y = x - 7$. 6. **Union means we combine both shaded regions:** The solution includes all points that satisfy either inequality. **Final answer:** The graph is the union of the half-planes below $y = -x - 5$ and above $y = x - 7$.