14. Reason Abstractly Compare and contrast the solutions $$ x

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14. Reason Abstractly Compare and contrast the solutions $$ x<2 $$ and $$ x \leq 2 $$.

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1. **Interpretation of $$ x < 2 $$**
   - This inequality represents all numbers $$ x $$ that are strictly less than 2.
   - The number 2 itself is not included in the solution set.
   - In interval notation, this is expressed as $$ (-\infty, 2) $$.

2. **Interpretation of $$ x \leq 2 $$**
   - This inequality represents all numbers $$ x $$ that are less than or equal to 2.
   - Here, the number 2 is included in the solution set.
   - In interval notation, this is expressed as $$ (-\infty, 2] $$.

3. **Comparison**
   - Both inequalities include all numbers that are less than 2.
   - The key difference is the endpoint:
     - $$ x < 2 $$ does **not** include 2.
     - $$ x \leq 2 $$ **does** include 2.

4. **Graphical Representation**
   - For $$ x < 2 $$:
     - The number 2 is represented by an open circle (or parenthesis) indicating that 2 is not a part of the solution.
   - For $$ x \leq 2 $$:
     - The number 2 is represented by a closed circle, indicating that 2 is included in the solution.

5. **Abstract Reasoning**
   - The difference between the two solution sets is precisely in the boundary value. Including or excluding a single point (in this case, 2) can lead to different conclusions in problems such as those involving limits, continuity, or optimization.
   - Even though the majority of the numbers in both sets are the same (all numbers less than 2), the inclusion of the boundary point in $$ x \leq 2 $$ can have meaningful implications depending on the context in which the inequality is used.

Thus, while both $$ x < 2 $$ and $$ x \leq 2 $$ seemingly cover almost the same numbers, the inclusion of 2 in the latter makes it distinct from the former.
$$ x < 2 $$ includes all numbers less than 2, excluding 2, while $$ x \leq 2 $$ includes all numbers less than or equal to 2, including 2.

14. Reason Abstractly Compare and contrast the solutions \( x<2 \) and \( x \leq 2 \).

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