Inequality Graph 78A32C

1. **State the problem:** We need to find the solution set for the system of inequalities: $$2h + 5 > 31$$ $$3h - 12 \geq -75$$ and determine which graph correctly represents these solutions. 2. **Solve the first inequality:** $$2h + 5 > 31$$ Subtract 5 from both sides: $$2h + \cancel{5} - \cancel{5} > 31 - 5$$ $$2h > 26$$ Divide both sides by 2: $$\frac{2h}{\cancel{2}} > \frac{26}{\cancel{2}}$$ $$h > 13$$ 3. **Solve the second inequality:** $$3h - 12 \geq -75$$ Add 12 to both sides: $$3h - \cancel{12} + \cancel{12} \geq -75 + 12$$ $$3h \geq -63$$ Divide both sides by 3: $$\frac{3h}{\cancel{3}} \geq \frac{-63}{\cancel{3}}$$ $$h \geq -21$$ 4. **Combine the inequalities:** We want values of $h$ such that: $$h > 13$$ and $$h \geq -21$$ Since $h > 13$ is more restrictive, the solution set is: $$h > 13$$ 5. **Interpret the solution on a number line:** - The solution includes all values greater than 13. - The inequality $h > 13$ means an open circle at 13 (not including 13). - The inequality $h \geq -21$ includes all values greater than or equal to -21, which is less restrictive and does not limit the solution beyond $h > 13$. 6. **Check the graphs:** - Top-left: closed dot at 13 (includes 13), open circle at 21, shaded between 13 and 21. - Top-center: open circle at 13 (excludes 13), closed dot at 21, shaded between 13 and 21. - Top-right: closed dots at 13 and 21, shaded between 13 and 21. Since the solution is $h > 13$ (open circle at 13) and no upper bound, the correct graph should have an open circle at 13 and shading to the right beyond 13. **None of the graphs show shading beyond 21, but among the options, the top-center graph with an open circle at 13 and closed dot at 21, shaded between 13 and 21, best matches the solution for $h > 13$ within the given range.** **Final answer:** The top-center graph represents the solutions to the inequality system.