Math Problem Statement
Solve the inequality in terms of intervals. (Enter your answer using interval notation.)
1\x < 2
Solution
To solve the inequality , follow these steps:
Step 1: Multiply both sides of the inequality by
We need to handle the inequality by isolating . However, be cautious when multiplying or dividing by , as it affects the direction of the inequality depending on whether is positive or negative.
So, we will break the solution into two cases:
Case 1:
When , multiplying by does not change the inequality direction. So, the inequality becomes:
Now, solve for :
This simplifies to:
Case 2:
When , multiplying by flips the inequality direction. So, the inequality becomes:
Solving for :
This simplifies to:
Step 2: Combine the results
We now combine the results of both cases:
- For , we have .
- For , we have .
Thus, the solution in interval notation is:
Final Answer:
The solution in interval notation is:
Would you like further details or have any specific questions on the solution?
Related Questions:
- How would the solution change if the inequality was ?
- What happens if the inequality is ?
- Can we solve this inequality without considering different cases for ?
- How would the solution change if we had ?
- What are the restrictions on for the expression to be defined?
Tip:
Always be cautious when multiplying or dividing by a variable in inequalities, as it may flip the inequality depending on the sign of the variable.
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Math Problem Analysis
Mathematical Concepts
Inequalities
Algebra
Interval Notation
Formulas
1/x < 2
Inequality multiplication rule for x > 0 and x < 0
Theorems
Multiplying or dividing by a variable in inequalities requires case analysis based on the sign of the variable.
Suitable Grade Level
Grades 9-11