Solve \( (x-2<5) \cup(x+7>6) \). \( \{x \mid-1

Ask by Martin Coles. in the United States

Feb 26,2025

To solve the compound inequality \( (x-2<5) \cup (x+7>6) \), let's break it down into its components. 1. For \( x-2 < 5 \): \[ x < 5 + 2 \implies x < 7 \] 2. For \( x+7 > 6 \): \[ x > 6 - 7 \implies x > -1 \] Now, we combine the two results using the union of intervals \( (-1, 7) \). Therefore, the solution set is: \[ \{x \mid -1 < x < 7\} \] About the real-world application, inequalities are prevalent in various fields. For example, engineers use them to determine acceptable ranges of materials' stress levels; scientists apply inequalities in data analysis to define thresholds for notable experiments or phenomena. Mastering inequalities equips you with essential skills for real-life problem-solving! When solving inequalities, it's easy to make mistakes by flipping the signs or misapplying the union and intersection rules. A common pitfall is neglecting to check whether the boundary points belong to the solution set. Always graph these inequalities to visualize the solution and double-check your work!