Quadratic Inequalities: Solving and Graphing

What are the steps to solve a quadratic inequality algebraically?

For the inequality $y > x^2 - 2x - 3$, what are the x-intercepts?

What does a solid line represent when graphing a quadratic inequality?

In the inequality $(x-3)(x+1) > 0$, what is the correct solution for x?

What is the vertex of the parabola in the inequality $(x-3)(x+1) > 0$?

How do we shade the region when graphing a quadratic inequality with $y < x^2 + 4x + 4$?

What does it mean when an inequality symbol is followed by an equal sign in a quadratic inequality?

If the inequality $(x-5)(x+2) eq 0$, what are the possible values for x?

In solving quadratic inequalities algebraically, what is the main reason for finding the x-intercepts?

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Quadratic Inequalities

Quadratic inequalities are mathematical expressions that involve a quadratic equation with an inequality symbol, such as "<" or ">". They are used to represent a range of values for the quadratic function. Solving quadratic inequalities algebraically and graphing them can help us understand the behavior of the function in different regions.

Solving Quadratic Inequalities Algebraically

To solve a quadratic inequality algebraically, we follow these steps:

Factor the quadratic equation.
Set each factor equal to zero to find the x-intercepts.
Find the vertex of the parabola.
Determine the region where the inequality holds true.

For example, consider the inequality:

$$ y > x^2 - 2x - 3 $$

Factor the quadratic equation:

$$ (x-3)(x+1) > 0 $$

Set each factor equal to zero:

$$ x-3 = 0 \Rightarrow x = 3 $$ $$ x+1 = 0 \Rightarrow x = -1 $$

Find the vertex of the parabola:

$$ (x-3)(x+1) > 0 \Rightarrow x > 3 \text{ or } x < -1 $$

The vertex is at x = 3, where y = 3.

Determine the region where the inequality holds true:

The inequality holds true for x > 3 or x < -1.

Graphing Quadratic Inequalities

To graph a quadratic inequality, we first graph the quadratic function associated with the inequality. The parabola can be either a solid line for ≤ or ≥, or a dotted line for < or >. Then, we shade the region above or below the parabola depending on the inequality symbol.

For example, the inequality:

$$ y < x^2 - 2x - 3 $$

will result in a parabola with a dotted line, and we shade the region below the curve.

Applications of Quadratic Inequalities

Quadratic inequalities have various applications in mathematics and other fields. For example, they can be used to represent the range of values for a quadratic function, or they can be used to solve optimization problems. In physics, quadratic inequalities can describe the behavior of certain systems, such as the motion of a particle in a potential field.

In conclusion, quadratic inequalities are a powerful tool for understanding the behavior of quadratic functions. By solving them algebraically and graphing them, we can gain valuable insights into the function's properties and applications.

Explore the algebraic methods and graphical representations of quadratic inequalities. Learn how to solve quadratic inequalities by factoring, finding x-intercepts, determining the vertex, and identifying the valid regions. Discover how to graph quadratic inequalities by plotting the associated parabola and shading the correct regions based on the inequality symbol.