Solving Quadratic Inequalities

Summary

This video tutorial walks through the process of solving quadratic inequalities. It covers key concepts such as factorizing quadratic expressions, identifying critical points where the equation equals zero, and interpreting the graph to determine the regions where the inequality holds true. The instructor explains how to work with inequalities like less than, greater than, and their equal counterparts, showing how to apply these methods to various examples. Key focus areas include graph interpretation, factorization, and determining the appropriate solution ranges for different inequality signs.

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Q & A

What is the first step in solving quadratic inequalities?

The first step is the same as solving a normal quadratic equation. You factorize the quadratic expression into two binomials, if possible.

Why is factorization considered the easiest method for solving quadratic inequalities?

Factorization is the easiest method because it allows us to find the roots (x-values) quickly, which are the critical points that help us determine the solution to the inequality.

How do you find the critical points of a quadratic inequality?

To find the critical points, set the quadratic equation equal to zero and solve for x. These are the points where the graph crosses the x-axis.

What does it mean for a quadratic inequality to be 'less than zero'?

When a quadratic inequality is 'less than zero', it means we are looking for the values of x where the graph of the quadratic is below the x-axis.

How can we identify the region where a quadratic inequality is 'greater than or equal to zero'?

For a quadratic inequality that is 'greater than or equal to zero', we look for the part of the graph that is above or on the x-axis, which corresponds to the region where the quadratic expression is positive or equal to zero.

What is the difference between 'greater than' (>) and 'greater than or equal to' (≥) in a quadratic inequality?

In a quadratic inequality, 'greater than' (>) means the graph is strictly above the x-axis, excluding the roots, while 'greater than or equal to' (≥) includes the roots where the quadratic equals zero.

What does the graph of a quadratic inequality look like when it is 'less than zero'?

When a quadratic inequality is 'less than zero', the graph will be below the x-axis, and we are interested in the x-values where the graph is in this region.

How do you handle a quadratic inequality with 'strictly less than' (<) or 'strictly greater than' (>) signs?

For 'strictly less than' (<) or 'strictly greater than' (>), the critical points (where the quadratic equals zero) are excluded from the solution. You only consider the intervals where the graph is either strictly below or strictly above the x-axis.

What is the solution to the inequality x² + 2x - 15 ≤ 0?

The solution is the interval from x = -5 to x = 3, i.e., -5 ≤ x ≤ 3. This corresponds to the region where the graph is either on or below the x-axis.

For the inequality x² - 6x + 8 > 0, what is the solution?

The solution is x < 2 or x > 4. This corresponds to the regions where the graph is strictly above the x-axis, excluding the points x = 2 and x = 4.