Demana Precalculus 8e — Section 2.7 Study Notes
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Section 2.7: Solving Equations in One Variable
Demana, Waits, Foley, Kennedy — Precalculus: Graphical, Numerical, Algebraic, 8th Edition
Big picture: this section applies everything from §2.1–2.6 to solving equations that involve rational
expressions — both algebraically (clearing fractions) and graphically (finding intersections) — with
special attention to a pitfall unique to this equation type: extraneous solutions.
Solving Rational Equations
Rational Equations
An equation involving rational expressions can always be written as
f (x)
= 0. If f, g share no
g(x)
common factors, the solutions are exactly the zeros of f (x).
Standard method: find the LCD (least common denominator) of every term, then multiply both
sides by the LCD to clear all fractions — reducing the problem to a polynomial equation solvable by
earlier methods (factoring, quadratic formula, etc.).
Extraneous Solutions
Definition — Extraneous Solution
When both sides of an equation are multiplied or divided by an expression containing the variable,
the resulting equation may gain solutions that are not solutions of the original equation. These are
extraneous solutions.
Why this happens & how to catch it
Multiplying by the LCD is valid only where the LCD is nonzero — but the algebra doesn’t ”know”
that, so it can introduce roots at exactly the values that make the LCD (or an original denominator)
zero. Always check every solution in the original equation — a value that makes any original
denominator zero is automatically extraneous, no arithmetic check needed.
ˆ It’s possible for every apparent solution to be extraneous — meaning the original equation has
no solution at all, even though the cleared-fractions equation does.
ˆ Graphical support matters here too: a solution that’s genuinely valid will show up as an actual
x-intercept of y = f (x) (the original expression moved to one side); an extraneous ”solution” will
not, because the function is undefined there.
Applications
Solving equations graphically often means finding where two curves intersect — move everything to
one side to solve f (x) = 0 directly, or graph the two original sides y1 = f (x) and y2 = g(x) separately
and find where y1 = y2 .
(80, 0.75)
f (x)
g(x) = 0.75
Demana Precalculus 8e — Section 2.7 Study Notes
mixture problem: solve
2
x + 17.5
x + 50
= 0.75 by intersection
Optimization problems (minimum/maximum) built from a rational function typically have no simple
algebraic shortcut — solve graphically, using a grapher’s minimum/maximum finder.
min ≈ (14.14, 56.57)
minimizing P (x) = 2x +
400
x
Worked-Example Techniques to Remember
ˆ Clearing fractions: identify the LCD by factoring every denominator first — this reveals shared
factors and keeps the LCD as small as possible.
ˆ Checking for extraneous solutions: substitute each candidate back into the original equation (not
the cleared version); any value making an original denominator zero is automatically rejected
without further calculation.
ˆ Recognizing a ”no solution” case: if every candidate from the cleared equation turns out extraneous, the correct conclusion is that the original equation has no solution — not that you made
an error.
ˆ Setting up a mixture/rate word problem: write the defining ratio as a word equation first (e.g.,
part
whole = concentration), then substitute expressions for each unknown quantity before solving.
ˆ Setting up an optimization word problem: express the quantity to minimize/maximize as a
single-variable function using a constraint equation to eliminate the second variable, then graph
and use a minimum/maximum finder.
ˆ Multiple solutions in applied problems: an equation solved graphically may have more than one
intersection point (as in the juice-can example) — check whether all mathematical solutions are
physically meaningful before reporting a final answer.
Source: Demana, Waits, Foley, Kennedy, Precalculus: Graphical, Numerical, Algebraic, 8th Edition, Pearson — Section
2.7 (pp. 228–232). Figures are original illustrations built for these notes, not reproductions of the textbook’s figures.