Lesson 8

1. Lesson 8

1.8. Explore 4

Mathematics 20-1 Module 4

Module 4: Quadratic Equations and Inequalities

Sign Analysis

Another algebraic approach is based on analyzing the signs of the factors in the quadratic expression.

Try This 3

This is a play button that opens Sign Analysis: A Strategy for Solving Quadratic Inequalities.

Launch the Sign Analysis: A Strategy for Solving Quadratic Inequalities interactive lesson to see how this method works. Keep in mind that this strategy is really a variation of the roots and test points method. As you work through the lesson, pay attention to the similarities between these two methods.

Turn to “Example 2” on pages 480 and 481 in the textbook. This example shows how you can use each of the algebraic methods that you have learned so far to solve a quadratic inequality of the form ax2 + bx + c < 0 where a < 0. You will benefit from reviewing both methods, but you will also find it helpful to see how a negative leading coefficient, when a < 0, is handled. As you work through the example, answer the following questions:

How could you choose test points so that it would be easier to evaluate the sign of the function?
How are negative constant factors dealt with?

Case Analysis

A third approach that can be used to solve quadratic inequalities is carried out by first identifying the conditions under which a quadratic inequality is true. The conditions are identified by analyzing the possible sign of each factor. A number line can then be used to determine when those conditions will be satisfied.

Try This 4

This is a play button that opens Case Analysis: Another Strategy for Solving Quadratic Equations.

Launch the interactive lesson titled Case Analysis: Another Strategy for Solving Quadratic Equations.

Turn to “Example 1” on page 478 of the textbook. Work your way through Method 1 only. Ask yourself the following questions as you work through the method:

How does the solution process change when the inequality symbol is ≤ or ≥ compared to < or >?
How is the reasoning process that leads to the solution interval in the example similar to the reasoning process used in the Case Analysis interactive lesson?

Self-Check 3

Turn to page 485 of the textbook to practise solving quadratic inequalities using the sign analysis and case analysis methods. Complete questions 5.a., 5.c., 6.b., and 6.d. You may wish to save your rough work in your course folder for future reference. If you make a mistake, don’t worry! Just give it another try. Answer

Although the solution intervals are different, the reasoning is the same: the solution is where the arrows overlap.

When the symbol is ≤ or ≥ , the solution includes the roots.

Combine the constant negative factor with one of the factors of the quadratic equation.

Choose large negative and large positive test points, which will make it easier to get the appropriate signs.