The expression is equivalent to , where is a constant and . What is the value of ?
When two rational expressions are equivalent and one denominator is given in factored form, first rewrite the other denominator to match it by factoring out constants and negative signs. Once the denominators are identical (for allowed -values), the numerators must match, so you can read off the unknown constant directly. As an alternative on the SAT, you can sometimes plug in an easy value of (that does not make a denominator zero) into both expressions and solve for the unknown constant, but be careful with signs and arithmetic.
Hints
Match the denominators
You are told is equivalent to . Try to rewrite so that it includes the factor in it.
Factor out a negative and a 2
Notice looks like the reverse of . Write as , and then factor .
Compare the simplified form with b/(x − 5)
After rewriting , substitute it back into the fraction and simplify. You should get something of the form (constant). That constant must be .
Desmos Guide
Graph the original expression
In Desmos, enter the function f(x) = 20/(10 - 2x).
Graph the family of possible equivalent expressions
On a new line, enter g(x) = b/(x - 5). Desmos will create a slider for the parameter b.
Adjust b until the graphs match
Move the b slider until the graph of g(x) lies exactly on top of the graph of f(x) for all visible (except at , where they are undefined). The value of b at that point is the constant that makes the two expressions equivalent.
Step-by-step Explanation
Rewrite the denominator in terms of x − 5
We want the denominator to look like .
Factor by first factoring out a negative sign:
Now factor :
So,
Substitute the factored form into the fraction
Replace the denominator in the original expression with the factored form:
You can view this as a constant over :
So the expression has the form
Find the constant and match it to b
Compute the constant :
So the expression simplifies to
This is equivalent to , so the numerators must be equal. Therefore,