Math Problem Statement
Solution
The inequality given is:
Step-by-Step Solution:
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Isolate the fraction: We start by eliminating the fraction. Multiply both sides of the inequality by , but be cautious: if is negative, we need to reverse the inequality sign.
First, assume (so no sign change). Then:
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Simplify the inequality: Expand the right-hand side:
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Subtract 12 from both sides:
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Solve for :
Or equivalently:
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Consider the case where : If is negative, multiplying by it would flip the inequality sign. The inequality would become:
Simplifying this:
Subtracting 12 from both sides:
Dividing by 4:
Or:
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Combine both cases: From both cases, we get that from the first case and from the second. Therefore, the solution is:
Final Answer in Interval Notation:
Let me know if you need any further clarification or a detailed explanation of any of the steps!
Would you like more details or have any questions? Here are some related questions:
- What if the inequality was strict () instead of ?
- How do we handle inequalities with absolute values?
- What happens if the inequality involves a quadratic expression?
- Can this method be applied to systems of inequalities?
- How would the solution change if we had ?
Tip: Always check for cases when multiplying or dividing inequalities by variables, as the inequality sign may change depending on whether the variable is positive or negative.
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Math Problem Analysis
Mathematical Concepts
Inequalities
Rational Expressions
Formulas
Inequality rules: When multiplying/dividing by a negative, flip the inequality
Solving rational expressions: Cross-multiply considering positive and negative cases
Theorems
Multiplication rule for inequalities
Case analysis when solving inequalities involving rational expressions
Suitable Grade Level
Grades 9-11