Card 1 of 20
EasyStudy card 1
Prompt
What is the first step in solving a rational equation?
Answer
List values that make any original denominator zero.
MathChapter 14: Rational Expressions
Flashcards
Solving Rational Equations Flashcards
Twenty recall cards on restrictions, clearing denominators, proportions, checks, and extraneous values.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
Why multiply by the LCD?
Answer
It clears all denominators while preserving equality for allowed values.
Card 3 of 20
EasyStudy card 3
Prompt
Which terms receive the LCD multiplier?
Answer
Every term on both sides.
Card 4 of 20
EasyStudy card 4
Prompt
When is cross multiplication directly valid?
Answer
When one fraction equals one fraction.
Card 5 of 20
EasyStudy card 5
Prompt
What is an extraneous solution?
Answer
A transformed-equation candidate that fails the original equation or its domain.
Card 6 of 20
EasyStudy card 6
Prompt
Where should candidates be checked?
Answer
In the original equation and against original restrictions.
Card 7 of 20
EasyStudy card 7
Prompt
What if the only candidate is excluded?
Answer
The equation has no solution.
Card 8 of 20
MediumStudy card 8
Prompt
For divisor \(x+5\), which value is excluded?
Answer
\(x=-5\).
Card 9 of 20
MediumStudy card 9
Prompt
What is the LCD of \(x\) and \(x-3\)?
Answer
\(x(x-3)\).
Card 10 of 20
MediumStudy card 10
Prompt
Can denominators cancel across an added term?
Answer
No. Clear them by multiplying every term by the LCD.
Card 11 of 20
MediumStudy card 11
Prompt
What does cross multiplying \(A/B=C/D\) produce?
Answer
\(AD=BC\), provided \(B,D\ne0\).
Card 12 of 20
MediumStudy card 12
Prompt
Why record restrictions before simplifying?
Answer
A canceled denominator factor still excludes its zero.
Card 13 of 20
MediumStudy card 13
Prompt
What two checks must an answer pass?
Answer
Every original denominator is nonzero and both original sides are equal.
Card 14 of 20
MediumStudy card 14
Prompt
Does a cleared equation automatically have the same solution set?
Answer
Only after excluded or extraneous candidates are removed.
Card 15 of 20
MediumStudy card 15
Prompt
What can a contradiction after clearing denominators mean?
Answer
The original rational equation has no solution.
Card 16 of 20
MediumStudy card 16
Prompt
What can an identity after clearing denominators mean?
Answer
Every value in the original domain is a solution.
Card 17 of 20
HardStudy card 17
Prompt
Solve \(1/x=1/5\).
Answer
\(x=5\), which satisfies \(x\ne0\).
Card 18 of 20
HardStudy card 18
Prompt
Why is \(x=2\) rejected from \(x/(x-2)=2/(x-2)\)?
Answer
It makes the original denominator zero.
Card 19 of 20
HardStudy card 19
Prompt
How can a quadratic candidate arise from a rational equation?
Answer
Clearing denominators can produce products containing \(x\), leading to a quadratic.
Card 20 of 20
HardStudy card 20
Prompt
What is the safest four-word rational-equation routine?
Answer
Restrict, clear, solve, check.
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