Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
466
Proportions and Cross Multiplication
When two fractions are equal, “X” marks the solution
21.1 Apply cross multiplication to the proportion and generate an
equivalent statement.
Cross multiplying a proportion eliminates the fractions from the equation.
Multiply the numerator of one fraction by the denominator of the other and set
the results equal.
21.2 Solve the proportion for x: .
Cross multiply (as described in Problem 21.1) to eliminate fractions from the
equation.
21.3 Solve the proportion for x: .
Cross multiply to eliminate fractions from the equation.
21.4 Solve the proportion for y: .
Cross multiply to eliminate fractions from the equation.
An equation
that states
two fractions are
equal is called a
proportion.
Distribute
3 to the
terms inside the
parentheses: 3(x – 3) =
3(x) + 3(–3) = 3x – 9.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
467
21.5 Solve the proportion for x: .
Cross multiply to eliminate fractions from the equation.
21.6 Solve the proportion for x: .
Cross multiply to eliminate fractions from the equation.
21.7 Solve the equation for x: .
Add to both sides of the equation to create a proportion.
Cross multiply to eliminate fractions from the proportion.
Subtract
x
2
from both
sides of the
equation.
Unlike
Problem
21.6, the x
2
-
terms won’t go
away, which means
you end up with a
quadratic equation.
Move all the terms to
the left side of the
equation so that
the right side
equals zero.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
468
Apply the quadratic formula.
Reduce the fraction to lowest terms.
21.8 Solve the proportion for x:
Cross multiply to eliminate fractions from the equation.
Distribute x and
then distribute 1 to each term of
x
2
– x – 2:
.
Chapter Twenty-One — Rational Equations and Inequalities
The Humongous Book of Algebra Problems
469
Note: Problems 21.9–21.10 refer to the equation .
21.9 Combine the expressions on the left side of the equation.
The least common denominator of and is x – 1. Write the fractions
using the least common denominator and simplify the expression.
Note: Problems 21.9–21.10 refer to the equation .
21.10 Solve the equation for x.
Substitute into the equation.
Cross multiply to eliminate fractions from the equation.
Solve the quadratic by factoring.
The solution to the equation is x = –2 or x = –1.
Replace
the left side
of the equation
with the sum
calculated in
Problem 21.9.
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