Two statements are given followed by four conclusions. Determine which conclusion is definitely true based on the statements. **Statements:** $$A D$$ $$C E \geq F = G$$ **Conclusions:** 1. $A G$

Question

Two statements are given followed by four conclusions. Determine which conclusion is definitely true based on the statements. **Statements:** $$A < B \geq C > D$$ $$C > E \geq F = G$$ **Conclusions:** 1. $A < E$ 2. $D < F$ 3. $G > C$ 4. $B > G$

Accepted Answer

**Solution:** 1. Analyze the given inequalities: - From the first statement: $$A < B \geq C > D$$ - From the second statement: $$C > E \geq F = G$$ 2. Check each conclusion: - **Conclusion 1: $A < E$** - We know $A < B$ and $B \geq C$, but there is no direct comparison between $A$ and $E$. - Since $E$ is related to $C$ by $C > E$, and $A$ is less than $B$ which is at least $C$, the relation between $A$ and $E$ cannot be definitively established. - **Conclusion 2: $D < F$** - From the first statement, $D$ is less than $C$. - From the second, $F$ is less than or equal to $E$, which is less than $C$. - No direct comparison between $D$ and $F$ is given, so this conclusion cannot be confirmed. - **Conclusion 3: $G > C$** - Given $F = G$ and $C > E \geq F$, it follows that $G \leq C$. - Hence, $G > C$ is false. - **Conclusion 4: $B > G$** - Since $B \geq C$ and $C > E \geq F = G$, it implies $B \geq C > G$. - Therefore, $B > G$ is definitely true. **Answer:** $B > G$

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