Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
470
Solving Rational Equations
Ditch the fractions or cross multiply to solve
Note: Problems 21.11–21.12 present different methods to solve the equation .
21.11 Apply cross multiplication to solve the proportion.
Apply the cross multiplication algorithm defined in Problem 21.1.
Note: Problems 21.11–21.12 present different methods to solve the equation .
21.12 Eliminate fractions by multiplying the entire equation by the least common
denominator and solve.
Multiply each term of the equation by 7(5x – 1), the least common denominator.
Distribute 4 on the right side of the equation and solve for x.
The solution to the equation verifies the solution in Problem 21.11.
This is
the method used
in Problems 21.2–
21.10.
This is
the strategy
you’ll use to solve
rational equations
that aren’t simple
proportions. When cross
multiplication won’t
do the trick, eliminate
the fractions using
the least common
denominator and
go from there.
You’re not
trying to get common
denominators, you’re
multiplying by the least
common denominator. In other
words, you’re multiplying each
term by
,
NOT
.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
471
21.13 Verify the solution to the equation from Problem 21.2 by multiplying
the entire equation by the least common denominator and solving for x.
Multiply both fractions in the proportion by 2
2
= 4, the least common
denominator.
The solutions to Problems 21.2 and 21.13 are equal , so both methods
of eliminating the fractions in a rational equation (multiplying by the least
common denominator and applying cross multiplication) produce the same
solution.
21.14 Solve the equation for x: .
Multiply the entire equation by 3x, the least common denominator.
Solve the equation for x.
15 = x
21.15 Solve the equation for x: .
Multiply the entire equation by x – 3, the least common denominator, to
eliminate the fraction.
Solve the quadratic equation by factoring.
Subtract x
2
from both sides of
the equation. That
leaves 15 on the left
side and x on the
right.
x – 3 is the
least common
denominator
because it is the
only denominator
(excluding the
implied denominators
of 1 in the terms
4 and 2x).
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
472
The solution to the equation is or x = 4.
21.16 Verify the solution to the equation from Problems 21.9–21.10
using the least common denominator to eliminate fractions before solving.
Multiply the entire equation by 2(x – 1), the least common denominator.
Solve the equation by factoring.
This solution matches the solution in Problem 21.10: x = –2 or x = –1.
21.17 Verify the solution(s) to the equation from Problem 21.7
using the least common denominator to eliminate fractions before solving.
Multiply the entire equation by (x + 4)(x – 1), the least common denominator.
Distribute –1 through the second quantity and simplify.
Add 12 to,
|and subtract
5x from, both
sides of the
equation so that only
zero appears left of
the equal sign. Then,
ip-op the sides of
the equation so
zero appears on
the right side
instead.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
473
Apply the quadratic formula.
Note that the solution verifies the solution to Problem 21.7.
21.18 Solve the equation for x: .
Factor the quadratic denominator: x
2
– 12x + 20 = (x – 10)(x – 2).
Multiply the entire equation by (x – 10)(x – 2), the least common denominator,
to eliminate fractions.
Solve for x.
Here’s how to
simplify this expres-
sion:
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
474
21.19 Solve the equation for x: .
Factor the quadratic denominator: x
2
– 1 = (x + 1)(x – 1).
Multiply the entire equation by (x + 1)(x – 1), the least common denominator, to
eliminate fractions.
Solve for x.
There are no valid solutions to the equation.
21.20 Solve the equation for x: .
Multiply the entire equation by (x + 2)(x – 5), the least common denominator, to
eliminate the fractions.
Apply the quadratic formula.
It looks like
x = 1 is a solution,
but if you plug it into
the original equation
to check it, two of the
three denominators
become 0, and you’re
not allowed to
divide by zero.
Set the
equation
equal to 0 by
adding x
2
to, and
subtracting 3x
and 10 from, both
sides of the
equation.
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