# Solve Rational Inequalities - Tutorial

How to solve rational inequalities? A tutorial with examples and detailed solutions.

Important Review

The sign of a rational expression P/Q , where P and Q are polynomials, depends on the signs of P and Q. In turn the signs of P and Q depend on the zeros of P and Q respectively if there are any. Hence the sign of P/Q depends on the zeros of both P and Q and changes (if it does!) only at these zeros. So to solve an inequality of the form P/Q > 0 (or P/Q < 0), we first find the zeros of P and Q then make a table of sign of P/Q.

Example 1 Solve the rational inequality given by

Solution to Example 1

We first find the zeros of the numerator and the denominator

The numerator is always negative and therefore does not change sign.

Zero of the denominator: solve -x + 4 = 0 to obtain x = 4.

The zero x = 4 divide the real number line into two intervals

(- , 4) , and (4 , + )

We now select and test values that are within each interval and test the rational expression to find its sign.

a) interval (- , 4)

test value x = 0

We now evaluate at x = 0 to find its sign.

(negative).

b) interval (4 , )

test value x = 5

We now evaluate at x = 5 to find its sign.

(positive).

We now put all the above results in a table

 -∞ 4 -∞ - undefined +

Conclusion The solution set of the given rational inequality is given by the interval

(4 , + )

Graphical solution to the given inequality.

Below is shown the graph fo the function y = (-3)/(- x + 4). It is easy to check graphically that y = (-3)/(- x + 4) is positive over the interval
(4 , ).

Example 2 Solve the rational inequality given by

Solution to Example 2

We first find the zeros of the numerator and the denominator

Zero of the numerator: solve x - 1 = 0 to obtain x = 1.

Zero of the denominator: solve x + 2 = 0 to obtain x = -2.

The two zeros x = 1 and x = -2 divide the real number line into three intervals (Note that the zeros are ordered from smallets to largest).

(- , -2) , (-2 , 1) and (1 , + )

We now select and test values that are within each interval and test the rational expression to find its sign.

a) interval (- , -2)

test value x = -3

We now evaluate at x = -3 to find its sign.

b) interval (-2 , 1)

test value x = 0

We now evaluate at x = 0 to find its sign.

c) interval (1 , + )

test value x = 2

We now evaluate at x = 2 to find its sign.

Let us now put all the above results in a table
In the interval (- , -2) , is positive
In the interval (- 2 , 1) , is negative
In the interval (1 , + ) , is positive

 -∞ -2 1 ∞ + undefined - 0 +

Conclusion The solution set of the given rational inequality is given by the interval

(- , -2) U (1 , + )

Graphical solution to the given inequality.

Below is shown the graph fo the function y = (x-1)/(x+2). It is easy to check graphically that y = (x-1)/(x+2) is positive over the interval
(- , -2) U (1 , + ).

Example 3  Solve the rational inequality given by

Solution to Example 3

NOTE: Do not multiply both sides by the LCD (x - 3)(x + 4) as you would do if this was an equation. The sign of (x - 3)(x + 4) changes with x and we do not know if the order of the inequality is to be changed or not.

Given

Write the inequality with the right side equal to zero by subtracting
3 / (x + 4) from both sides.

Rewrite the inequality so that the two terms making the left side have common denominator.

Multiply factors, add the two rational expressions on the left side of the inequality and group like terms in the numerator to obtain

The zero of the numerator: - x + 17 = 0 is x = 17
The zeros of the denominator: (x - 3)(x + 4) = 0 are x = 3 and x = -4

The three zeros divide the real number line into four intervals (Note that the zeros are ordered from the smallest to the largest).

(- , -4) , (-4 , 3) , ( 3 , 17) , (17 , +)

a) (- , -4) test value: x = -5
We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = -5 to find its sign.
( -x + 17) / [ (x - 3)(x + 4) ]
= ( -(-5) + 17) / [ (-5 - 3)(-5 + 4) ]
= 22/8 (positive)

b) (- 4 , 3) test value: x = 0
We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 0 to find its sign.
( -x + 17) / [ (x - 3)(x + 4) ]
= ( 0 + 17) / [ (0 - 3)(0 + 4) ]
= -17 / 12 (negative)

c) (3 , 17) test value: x = 4
We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 4 to find its sign.
( -x + 17) / [ (x - 3)(x + 4) ]
= (- 4 + 17) / [ (4 - 3)(4 + 4) ]
= 13/8 (positive)

d) (17 , +) test value: x = 18
We now evaluate ( -x + 17) / [ (x - 3)(x + 4) ] at x = 18 to find its sign.
( -x + 17) / [ (18 - 3)(x + 4) ]
= ( -18 + 17) / [ (18 - 3)(18 + 4) ]
= -1/330 (negative)

Let us now put all the above results in a table

 -∞ -4 3 17 -∞ + undefined - undefined + 0 -

Conclusion The solution set of the given rational inequality is given by the interval

(- 4 , 3) U (17 , + )

More references and links on how to Solve Equations, Systems of Equations and Inequalities.