Solving Rational Inequalities

Solving Rational Inequalities

How to solve rational inequalities problems: examples and their solutions.

Example 1

Solve the given inequality. 1/x - 1/2x >= 3

Write the excluded value:
x ≠ 0

Excluded value (extraneous solution)

Move all the terms to the left side.

To add and subtract the fractions,
make each fraction's denominator to the LCD.

Multiply the missing factors
to both of the numerator and the denominator.

Adding and subtracting rational expressions

Multiply both sides to -1
to make the coefficient of x term (+).

This changes the order of the inequality sign.

Divide both sides by 2.

This doesn't change the order of the inequality sign.

Multiply the square of the denominator, x2,
to both of the numerator and the denominator.

x ≠ 0 and x2 ≥ 0.
So x2 > 0.

This doesn't change the order of the inequality sign.

By doing this,
the rational inequality becomes a quadratic inequality.
So you can solve this.

Solving quadratic inequalities

The zeros of x(6x - 1) are 0 and 1/6.

Mark the zeros on the number line.

Draw an empty circle on x = 0 (excluded value)
and a full circle on x = 1/6.

Draw y = x(6x - 1)
using x = 0, 1/6.

The left side is lesser than 0.
(See the most recently changed inequality sign. blue)

So draw the region on the number line
where the graph is below the number line.

Then 0 < x ≤ 1/6.

Example 2

Solve the given inequality. 4/(x - 1) + 1 <= 1/x

Write the excluded values:
x ≠ 1, x ≠ 0

Excluded value (extraneous solution)

Move all the terms to the left side.

To add and subtract the fractions,
make each fraction's denominator to the LCD.

Multiply the missing factors
to both of the numerator and the denominator.

Adding and subtracting rational expressions

Factoring a perfect square trinomial

Multiply the square of the denominator, [x(x - 1)]2,
to both of the numerator and the denominator.

x ≠ 0, x ≠ 1.
And [x(x - 1)]2 ≥ 0.
So [x(x - 1)]2 > 0.

This doesn't change the order of the inequality sign.

By doing this,
the rational inequality becomes a polynomial inequality.
So you can solve this.

Solving polynomial inequalities

The zeros of x(x - 1)(x + 1)2 are 0, 1, and -1.

Mark the zeros on the number line.

Draw a full circle on x = -1
and an empty circle on x = 0, 1 (excluded values).

Draw y = x(x - 1)(x + 1)2.

The left side is lesser than 0.

So draw the region on the number line
where the graph is below the number line.

Then x = -1 or 0 < x < 1.