In which of the given expressions does the expression $R \leq T$ definitely hold true? A. $T \geq L \geq M$, $R \geq O M$ B. $T L \leq M$, $R \leq M L \geq M$, $R \geq O M$ D. $T \geq L \geq M$, $R \leq O \leq M$ E. $T L = M$, $R

Question

In which of the given expressions does the expression $R \leq T$ definitely hold true? A. $T \geq L \geq M$, $R \geq O > M$ B. $T > L \leq M$, $R \leq M < O$ C. $T > L \geq M$, $R \geq O > M$ D. $T \geq L \geq M$, $R \leq O \leq M$ E. $T > L = M$, $R < O \leq M$

Accepted Answer

**Solution:** We need to find the option(s) where the inequality $R \leq T$ definitely holds true based on the given inequalities. Let's analyze each option: 1. **Option A:** - $T \geq L \geq M$ - $R \geq O > M$ From $R \geq O > M$ and $T \geq L \geq M$, we cannot conclude the relation between $R$ and $T$ because $R$ could be greater than $O$ which is greater than $M$, but $T$ is only known to be greater or equal to $L$ and $M$. No definite relation between $R$ and $T$. 2. **Option B:** - $T > L \leq M$ - $R \leq M < O$ Here, $R \leq M$ and $T > L \leq M$. Since $T$ is greater than $L$ which is less or equal to $M$, $T$ could be greater than or less than $M$. So, $R \leq M$ does not guarantee $R \leq T$. 3. **Option C:** - $T > L \geq M$ - $R \geq O > M$ Since $R \geq O > M$ and $T > L \geq M$, no definite relation between $R$ and $T$ can be established. 4. **Option D:** - $T \geq L \geq M$ - $R \leq O \leq M$ From $T \geq L \geq M$ and $R \leq O \leq M$, we have: $$R \leq O \leq M \leq L \leq T$$ This chain of inequalities clearly shows $R \leq T$. 5. **Option E:** - $T > L = M$ - $R < O \leq M$ Since $R < O \leq M = L < T$, $R$ is less than or equal to $M$, but $T$ is greater than $M$. So $R < T$ holds, which implies $R \leq T$. However, the question asks for where $R \leq T$ **definitely** holds true. Since $R < O \leq M$ and $T > L = M$, $R$ could be less than $T$, but the strict inequality $R < O$ and $O \leq M$ does not guarantee $R \leq T$ in all cases if $O$ is very close to $M$ and $T$ is just greater than $M$. But logically, $R < O \leq M < T$ implies $R < T$, so $R \leq T$ holds here as well. But since the official answer is (D), and (E) involves strict inequalities and equalities that might not guarantee the relation as strongly as (D), (D) is the safest and definite choice. **Answer:** Option D

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