Math Problem Statement
Solve this Rational Inequality (x + 12)/(x + 2) ≥ 2
Solution
We are tasked with solving the rational inequality:
Step 1: Rearrange the inequality
First, subtract 2 from both sides to bring all terms to one side:
Next, express with the same denominator as :
Simplify the right-hand side:
Distribute and simplify the numerator:
Step 2: Find critical points
The inequality changes sign at the points where the numerator or denominator equals zero.
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Set the numerator :
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Set the denominator :
So the critical points are and . These values divide the real number line into three intervals: , , and .
Step 3: Test intervals
We now test points from each interval in the expression :
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Interval : Choose :
Since is less than zero, this interval does not satisfy the inequality.
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Interval : Choose :
Since 4 is greater than zero, this interval satisfies the inequality.
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Interval : Choose :
Since is less than zero, this interval does not satisfy the inequality.
Step 4: Consider boundary points
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At , the expression becomes zero: Since we are looking for values where the expression is greater than or equal to zero, is included in the solution.
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At , the expression is undefined because the denominator is zero. So, is not included in the solution.
Final Solution
The solution to the inequality is:
Do you need further clarification or details on any of these steps?
Related Questions:
- How do you solve other rational inequalities with different denominators?
- What is the difference between strict inequalities and non-strict inequalities?
- How do asymptotes affect the solution of rational inequalities?
- How can we check for extraneous solutions in inequalities?
- How do we handle rational inequalities involving higher degree polynomials?
Tip:
Always check the critical points by plugging them back into the original inequality to avoid including undefined values or missing valid boundary solutions.
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Math Problem Analysis
Mathematical Concepts
Algebra
Rational Inequalities
Critical Points
Sign Testing
Formulas
(x + 12)/(x + 2) ≥ 2
Critical points: numerator = 0, denominator ≠ 0
Theorems
Sign chart method
Inequalities involving rational functions
Suitable Grade Level
Grades 10-12
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