Which compound inequality has no solution?
Understand the Problem
The question is asking for an example of a compound inequality that cannot be satisfied by any value, meaning there is no solution that can fulfill the given conditions.
Answer
The example of a compound inequality that cannot be satisfied by any value is: $x < 2 \quad \text{and} \quad x > 5$.
Answer for screen readers
The example of a compound inequality that cannot be satisfied by any value is:
$$ x < 2 \quad \text{and} \quad x > 5 $$
Steps to Solve
- Identify the structure of a compound inequality
A compound inequality usually involves two parts and can be of two types: "and" or "or." "And" inequalities require that both conditions be true simultaneously, while "or" inequalities only need one condition to be true.
- Create an example of an unsolvable compound inequality
For an impossible scenario, we can create a compound inequality like this:
$$ x < 2 \quad \text{and} \quad x > 5 $$
Here, we are stating that $x$ must be less than 2 and greater than 5 at the same time.
- Analyze the compound inequality
Now, let's consider what this means. The first condition states that $x$ must be less than 2, and the second condition states that $x$ must be greater than 5.
- Conclusion of unsatisfiability
Since no number can be both less than 2 and greater than 5 simultaneously, this compound inequality has no solution.
The example of a compound inequality that cannot be satisfied by any value is:
$$ x < 2 \quad \text{and} \quad x > 5 $$
More Information
This type of compound inequality illustrates the concept of disjoint intervals on the number line. There is no overlap between the two conditions, making it impossible for any value to satisfy both at the same time.
Tips
- A common mistake is to misinterpret the meanings of “and” and “or” in compound inequalities. Remember, "and" requires both conditions to be true, while "or" allows for either condition to suffice.
- Another oversight involves creating compound inequalities that generate overlapping ranges; ensure to clearly define disjoint conditions to illustrate unsolvability.
