Explanation The boundary points separate the number line into different regions. And, all the values of the variable in that region satisfy the inequality that is greater than 0 or all the values of the variable in that region satisfies the inequality that is less than 0. To find the boundary points, replace the inequality sign with the equal sign and solve for any variable. And, these boundary points separate the number line into different regions. Then, evaluate the quadratic expression at a test point in each region. After testing, it can be concluded that all the values of the variable in that region satisfy the inequality that is greater than 0 or all the values of the variable in that region satisfies the inequality that is less than 0. Therefore, the boundary points separate the number line into different regions. Example: Consider an inequality. x 2 + x − 20 > 0 Replace the inequality sign with the equal sign and solve for x . x 2 + x − 20 = 0 x 2 + 5 x − 4 x − 20 = 0 x ( x + 5 ) − 4 ( x + 5 ) = 0 ( x + 5 ) ( x − 4 ) = 0 That is, x = − 5 and x = 4 So, the boundary points are − 5 and 4. These boundary points separate the number line into three regions. First region: x < − 5 Second region: − 5 < x < 4 Third region: x > 4 Now, evaluate the quadratic expression x 2 + x − 20 at a test point in each region. For the first region x < − 5 , consider a test point x = − 10

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