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 Tuesday 23rd of January

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 Dependent Variable

 Number of inequalities to solve: 23456789
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# Solving Quadratic Inequalities with a Sign Graph

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

Use a sign graph to solve the inequality x2 + 3x - 10 > 0.

Solution

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 = 0x + 5 > 0 x + 5 < 0 if x = -5if 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 = 0x - 2 > 0 x - 2 < 0 if x = 2if 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

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

Solution

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.