Math Problem Statement
Solution
Let's go through each of the problems step-by-step. I'll start with the first one:
1.
Step 1: Cross-multiply to eliminate the fractions: This simplifies to:
Step 2: Solve for :
Solution for Problem 1:
2.
Step 1: Multiply both sides by (assuming ):
Step 2: Distribute the 5 on the right-hand side:
Step 3: Bring all terms involving to one side and constants to the other side:
Step 4: Solve for :
Solution for Problem 2:
3.
Step 1: Combine the fractions on the left-hand side:
Step 2: Set this equal to the right-hand side:
Step 3: Cross-multiply:
Step 4: Solve for :
Solution for Problem 3:
4.
Step 1: Move all terms to one side:
Step 2: Combine into a single fraction:
Step 3: Solve the inequality . The critical points are at and .
Analyzing the sign of the expression around these points:
- For , the expression is positive.
- For , the expression is negative.
- For , the expression is positive.
So, the solution is (but since that makes the denominator zero).
Solution for Problem 4:
5.
Step 1: Recognize that , so the right-hand side simplifies to:
Step 2: Combine the left-hand side into a single fraction:
Step 3: Set this equal to the right-hand side:
Step 4: Cancel the denominators and solve:
Solution for Problem 5: or
Would you like more details on any step or have any other questions? Here are five questions to deepen your understanding:
- How do you determine the critical points in inequalities with rational expressions?
- Why is it important to check for excluded values in rational equations?
- How does cross-multiplying help in solving fractional equations?
- How can you simplify complex rational expressions before solving them?
- What is the significance of analyzing signs in rational inequalities?
Tip: When solving inequalities, always analyze the critical points carefully, and check which intervals satisfy the inequality.
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Math Problem Analysis
Mathematical Concepts
Algebra
Inequalities
Rational Equations
Cross-multiplication
Quadratic Expressions
Formulas
Cross-multiplication
Rational expressions simplification
Solving inequalities using critical points
Theorems
Cross-multiplication method
Solving inequalities by sign analysis
Suitable Grade Level
Grades 9-12