Solving Rational Inequalities with a Sign Graph
The inequalities
are called rational inequalities. When we solve equations that involve rational expressions,
we usually multiply each side by the LCD. However, if we multiply each
side of any inequality by a negative number, we must reverse the inequality, and
when we multiply by a positive number, we do not reverse the inequality. For this
reason we generally do not multiply inequalities by expressions involving variables.
The values of the expressions might be positive or negative. The next two examples
show how to use a sign graph to solve rational inequalities that have variables in the
denominator.
Example 1
Solving a rational inequality
Solve
and graph the solution set.
Solution
We do not multiply each side by x  3. Instead, subtract 2 from each side to get 0
on the right:

≤ 0 


≤ 0 
Get a common denominator. 

≤ 0 
Simplify. 

≤ 0 
Subtract the rational expressions. 

≤ 0 
The quotient of x + 8 and x  3 is
less than or equal to 0. 
Examine the signs of the numerator and denominator:
Make a sign graph as shown in the figure below.
Using the rule for dividing signed numbers
and the sign graph, we can identify where the quotient is negative or zero. The
solution set is (∞, 3)
Ãˆ [8, ∞). Note that 3 is not in the solution set because the
quotient is undefined if x = 3. The graph of the solution set is shown in the
next figure.
Helpful hint
By getting 0 on one side of the
inequality, we can use the
rules for dividing signed numbers.
The only way to obtain a
negative result is to divide
numbers with opposite signs.
Caution
Remember to reverse the inequality sign when multiplying or
dividing by a negative number. For example, x  3 > 0 is equivalent to x > 3. But
x + 8 > 0 is equivalent to x > 8, or x < 8.
