Given the statements: - $I \geq J = K

Question

Given the statements:

- $I \geq J = K < L$
- $M \leq J$
- $N < K$

Determine which of the following conclusions are true:

I. $M \leq I$

II. $N < L$

III. $I > L$

IV. $J = N$

Accepted Answer

**Solution:**

1. Given statements:
   - $I \geq J = K < L$
   - $M \leq J$
   - $N < K$

2. From $I \geq J = K$, we know $I \geq J$ and $J = K$.

3. From $M \leq J$ and $J = K$, it follows that $M \leq J = K$.

4. From $N < K$ and $J = K$, we have $N < K = J$.

5. Now, analyze each conclusion:

- **I. $M \leq I$**:
     Since $M \leq J$ and $I \geq J$, by transitivity, $M \leq I$ is true.

- **II. $N < L$**:
     Given $N < K$ and $K < L$, by transitivity, $N < L$ is true.

- **III. $I > L$**:
     Given $I \geq J = K < L$, $I$ is not necessarily greater than $L$. So this is false.

- **IV. $J = N$**:
     Given $N < K = J$, $N$ is less than $J$, so this is false.

**Answer:** Both I and II are true.

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