JOVANA
Library Glossary Getting Started Three Levels Fields How it works Mission
Join the mission
All guides

Rational Equations, Extraneous Roots, and Inverse Variation

Clear the denominators to turn a rational equation into an ordinary one — then always check, because clearing can manufacture solutions that don't really fit. Finish with inverse variation, a story rational expressions tell about the world.

Solving by clearing denominators

A rational equation sets rational expressions equal. The cleanest solving move is clearing the denominators: multiply every term on both sides by the LCD. The fractions vanish and you are left with a polynomial equation — often linear, sometimes a quadratic equation — that you already know how to solve.

Solve   3/(x - 2)  +  1  =  x/(x - 2)

LCD = (x - 2).  Multiply every term by (x - 2):
   3 + (x - 2) = x
   3 + x - 2 = x
   x + 1 = x
   1 = 0      <- false, no solution

There is NO value of x that works.
(And x = 2 was excluded from the start anyway.)
Multiplying every term by the LCD removes all fractions in one stroke.

Why you must check: extraneous solutions

Clearing denominators can introduce an extraneous solution — a value that solves the cleared polynomial but makes an original denominator zero, so it was never allowed. It is not a real solution; it is an artifact of the multiplication. The fix is checking each solution against the original equation and discarding any that hits an excluded value.

Solve   x/(x - 4)  =  4/(x - 4) + 2

LCD = (x - 4).  Multiply every term:
   x = 4 + 2(x - 4)
   x = 4 + 2x - 8
   x = 2x - 4
   -x = -4
   x = 4

CHECK x = 4 in the original:
   denominator (x - 4) = 0  ->  undefined!
   x = 4 is EXTRANEOUS, reject it.

Solution set: empty (no valid solution).
x = 4 solves the cleared line but is an excluded value — it is extraneous.

Inverse variation: a rational story

Inverse variation says two quantities have a constant product: y = k/x, equivalently xy = k. As one grows the other shrinks. The number k is the constant of proportionality, fixed for a given situation. Notice y = k/x is itself a rational expression, so x = 0 is excluded — which makes physical sense (you cannot divide the fixed product by nothing).

y varies inversely with x, and y = 6 when x = 4.

Model:   y = k/x
Find k:  6 = k/4  ->  k = 24
So:      y = 24/x   (equivalently xy = 24)

Predict y when x = 3:
   y = 24/3 = 8

As x falls from 4 to 3, y rises from 6 to 8 — the product xy stays 24.
Find k from one known pair, then the model predicts every other pair.

Contrast this with direct variation (y = kx), where the two quantities rise and fall together. Inverse variation is the rational cousin — and with it you have now seen rational expressions all the way from definition, through arithmetic, to solving and modeling.