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Rational (Math)

See how to simplify a rational expression/equation/inequality.
17 examples and their solutions.

Simplifying a Rational Expression

Example

x - 5x2 - 7x + 10
Solution

Example

2x2 + x - 1x2 - 2x - 3
Solution

Dividing Rational Expressions

Example

x3x - 1 ÷ x + 72x
Solution

Example

x2 - 4x - 1 ÷ x + 25
Solution

Complex Fraction

Formula

ab cd = adbc

Example

x + 2x 6x - 1
Solution

Example

4x - 1x (2x - 1)2
Solution

Adding and Subtracting Rational Expressions

Example

3x + xx + 2
Solution

Example

2xx2 - 1 - 5x2 + x
Solution

Excluded Value

Formula

ab

Excluded value: b = 0
The excluded value of a fraction
is the x value
that makes the denominator 0.
The denominator cannot be 0.
So the excluded value
cannot be the solution.
To find the excluded value,
set (denominator) = 0.

Example

x2 + 5x - 13x - 2
Solution

Rational Equation

Example

3x + xx - 1 = 1x(x - 1)
Solution

Rational Inequality

Example

4x - 1 + 1 ≤ 1x
Solution

Partial Fraction Decomposition

Partial fraction decomposition is a way
to split a rational expression
to its partial fractions.
(= fractions with reduced denominators)

Example

3x - 2x(x - 1)
Solution

Example

5x2 - 1x2(x + 1)
Solution

Formula

1AB = 1B - A(1A - 1B)
When B - A = (constant),
use this formula.

Example

1x(x + 1)
Solution

Rational Function: Graph

Graph: y = a/x (a > 0)

The numerator a is (+).
Then the graph is
on the right top and the left bottom
of the axes.

Graph: y = a/x (a < 0)

The numerator a is (-).
Then the graph is
on the left top and the right bottom
of the axes.

Asymptotes


y = ax
Asymptotes: x = 0
y = 0
A rational function y = a/x has two asymptotes:
a vertical asymptote
and a horizontal asymptote.
(asymptote: a line the graph follows)
To find the asymptotes of y = ax,
1. Set (denominator) = 0.
x = 0
2. Set (fraction) = 0.
→ y = 0

Example

Graph y = 4x - 1 + 2.
Solution

Example

Graph y = 13 - x - 1.
Solution

Example

Graph y = 3x - 5x - 2.
Solution