📚 Topic Summary
Rational equations involve fractions where the numerator and/or denominator contain variables. Solving them requires clearing the fractions, typically by multiplying both sides of the equation by the least common denominator (LCD). It's crucial to check your solutions because multiplying by an expression containing a variable can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one.
Extraneous solutions occur when a potential solution makes the denominator of the original rational equation equal to zero. Always substitute your solutions back into the original equation to verify they are valid.
🧠 Part A: Vocabulary
Match each term with its definition:
| Term |
Definition |
| 1. Rational Equation |
A. A solution that appears valid but does not satisfy the original equation. |
| 2. Least Common Denominator (LCD) |
B. A value that makes the denominator of a rational expression equal to zero. |
| 3. Extraneous Solution |
C. An equation containing at least one fraction whose numerator and/or denominator are polynomials. |
| 4. Solution Set |
D. The smallest multiple that all denominators divide into evenly. |
| 5. Restricted Value |
E. The set of all values that satisfy the equation. |
📝 Part B: Fill in the Blanks
To solve rational equations, first find the ______ (1) of all denominators. Multiply both sides of the equation by the ______(2) to eliminate the fractions. After solving the resulting equation, it is important to check for ______ (3) by substituting each potential solution back into the ______ (4) equation. If a value makes any denominator equal to zero, it is an ______ (5) solution and must be excluded from the solution set.
💡 Part C: Critical Thinking
Explain, in your own words, why it is necessary to check for extraneous solutions when solving rational equations. What happens if you don't check?