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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 ax2 + 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 x2 + 3x - 10 > 0.


Because the left-hand 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) È (2, ).

In the next example we will make the procedure from Example 1 a bit more efficient.

Example 2

Solving a quadratic inequality

Solve 2x2 + 5x 3 and graph the solution set.


Rewrite the inequality with 0 on one side:

2x2 + 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.

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