Solving Quadratic Inequalities with a Sign Graph
An inequality involving a quadratic polynomial is called a quadratic inequality.
Quadratic Inequality
A quadratic inequality is an inequality of the form
ax^{2} + bx + c > 0, where a, b, and c are real numbers with a
≠ 0. The inequality symbols <,
≤, and ≥ may also be used.
If we can factor a quadratic inequality, then the inequality can be solved with a
sign
graph, which shows where each factor is positive, negative, or zero.
Example 1
Solving a quadratic inequality
Use a sign graph to solve the inequality x^{2} + 3x  10 > 0.
Solution
Because the lefthand side can be factored, we can write the inequality as
(x + 5)(x  2) > 0.
This inequality says that the product of x + 5 and x  2 is positive. If both factors
are negative or both are positive, the product is positive. To analyze the signs of
each factor, we make a sign graph as follows. First consider the possible values of
the factor x + 5:
Value 
Where 
On the number line 
x + 5 = 0 x + 5 > 0
x + 5 < 0 
if x = 5 if x > 5
if x < 5 
Put a 0 above 5. Put + signs to the right of 5.
Put  signs to the left of 5. 
The sign graph shown in the figure below for the factor x + 5 is made from the information
in the preceding table.
Now consider the possible values of the factor x  2:
Value 
Where 
On the number line 
x  2 = 0 x  2 > 0
x  2 < 0 
if x = 2 if x > 2
if x < 2 
Put a 0 above 2. Put + signs to the right of 2.
Put  signs to the left of 2. 
We put the information for the factor x  2 on the sign graph for the factor x
+ 5
as shown in the figure below.
We can see from this figure that the product is positive if
x < 5 and the product is positive if x >2. The solution set for the quadratic
inequality is shown in the next figure.
Note that 5 and 2 are not included in the graph because
for those values of x the product is zero. The solution set is (∞,5)
Ãˆ (2, ∞).
In the next example we will make the procedure from Example 1 a bit more
efficient.
Example 2
Solving a quadratic inequality
Solve 2x^{2} + 5x
≤ 3 and graph the solution set.
Solution
Rewrite the inequality with 0 on one side:
2x^{2} + 5x  3 
≤ 0 

(2x  1)(x + 3) 
≤ 0 
Factor 
Examine the signs of each factor:
Make a sign graph as shown in the figure below.
The product of the factors is negative between 3 and
, when one factor is negative and the other is positive. The product
is 0 at 3 and at
. So the solution set is the interval
. The graph of the
solution set is shown in the following figure.
