Advanced Math · Rational Expressions
SAT Rational Expressions Practice Questions (Free + Explanations) | Quiz 6
For , if , what is the value of ?
Start with
Multiply both sides by to get . Then solve: , so , which gives . This value is allowed because .
To solve a rational equation with one fraction, multiply both sides by the denominator, then solve the resulting linear equation and check that the denominator is not zero.
Which expression is equivalent to for all values of where the expression is defined?
To divide by a fraction, multiply by its reciprocal:
The factor cancels, leaving
So the equivalent expression is .
To simplify a quotient of rational expressions, rewrite division as multiplication by the reciprocal, then cancel any common factors.
At a concert venue, the average ticket cost per person, in dollars, for a group of people is modeled by
where . On the graph of in the -plane, which point represents a group whose average ticket cost is dollars per person?
A point on the graph has the form , so we need . Solve:
Multiply both sides by :
So the point on the graph is .
For a graph of a rational function in context, interpret coordinates as input-output pairs: the first coordinate is the independent variable, and the second is the function value.
For nonzero numbers and , suppose
and
What is the value of ?
Use the identity
Since , the first equation becomes
Now solve for by multiplying both sides by :
So,
When a system includes and , rewrite the rational expression as to connect the two equations and solve for the product.
For all values of for which the expression is defined, the equation
is true. What is the value of ?
Rewrite the right-hand side as a single fraction:
Since this expression equals
for all , the numerators must be equal:
Subtract from both sides:
so
To compare rational expressions with the same denominator, rewrite all terms over that denominator and then match numerators when the expressions are equal for all allowed values of the variable.
Your results
Estimated SAT Math band
500-550
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practiceYour results
Estimated SAT Math band
500-550
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practiceYour results
Estimated SAT Math band
600-650
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practiceYour results
Estimated SAT Math band
600-650
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practiceYour results
Estimated SAT Math band
700+
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practiceYour results
Estimated SAT Math band
700+
Illustrative range from this short quiz—not an official College Board score.
Adaptive practice, weak-area review, and timed tests live in the MCQsLearn app—pick up where you left off on your phone.
More SAT Math practice