How to Solve Quadratic Inequality - Part 2

Summary

In this video, Teacher G walks viewers through solving a quadratic inequality, using the example of x^2 - x - 12 > 0. The first step is to convert the inequality into a quadratic equation, solve for critical points, and plot them on a number line. After evaluating, Teacher G demonstrates that values between -3 and 4 are not solutions. He explains the inclusion of critical points and shows how to express the solutions in interval notation. The video provides a clear, step-by-step explanation of solving quadratic inequalities, with an interactive approach encouraging viewer participation.

Takeaways

• 😀 The main goal of the video is to explain how to solve quadratic inequalities.
• 😀 The quadratic inequality to solve is: x^2 - x - 12 > 0.
• 😀 To start solving, you first turn the inequality into an equation by setting it equal to 0: x^2 - x - 12 = 0.
• 😀 The next step is to factor the quadratic equation, resulting in: (x - 4)(x + 3) = 0.
• 😀 The critical points of the equation are found by solving for x, which gives x = 4 and x = -3.
• 😀 A number line is used to visualize the intervals created by the critical points: (-∞, -3), (-3, 4), and (4, ∞).
• 😀 The next step involves testing values within each interval to check whether they satisfy the inequality.
• 😀 In the interval (-∞, -3), the inequality is satisfied because the test value yields a positive result.
• 😀 In the interval (-3, 4), the inequality is not satisfied because the test value yields a negative result.
• 😀 In the interval (4, ∞), the inequality is satisfied because the test value yields a positive result.
• 😀 The solution set of the quadratic inequality is the union of the intervals (-∞, -3) and (4, ∞), represented in interval notation as: (-∞, -3) ∪ (4, ∞).

Q & A

What is the main topic of the video?

The main topic of the video is solving quadratic inequalities.

What is the first step in solving a quadratic inequality?

The first step is to turn the inequality into a quadratic equation by changing the inequality sign to an equal sign.

What is the given quadratic inequality in the video?

The given quadratic inequality is x² - x - 12 > 0.

How do you solve the quadratic equation formed from the inequality?

To solve the quadratic equation, factor the expression x² - x - 12 to get (x - 4)(x + 3) = 0.

What are the critical points for the inequality?

The critical points are x = 4 and x = -3, which are found by solving (x - 4)(x + 3) = 0.

Why is it important to use a number line in this process?

The number line helps visualize the possible solution intervals and the critical points, which divide the line into different parts to test for solutions.

How do you check if the solution satisfies the inequality?

You plug in a test value from the intervals between the critical points into the factored inequality and check whether the result satisfies the inequality.

What was the result when zero was tested as a solution?

When zero was tested, it resulted in a false statement, meaning x = 0 is not a solution to the quadratic inequality.

Are the critical points included in the solution?

No, the critical points are not included in the solution. The critical points are represented by open circles on the number line.

How should the solution be expressed in interval notation?

The solution should be expressed as (-∞, -3) U (4, ∞), using parentheses to indicate that -3 and 4 are not included in the solution.