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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

When you finish watching this video lesson, you will be able to graph and solve your own quadratic inequality. Learn what steps you need to take and what to watch for.

A **quadratic inequality** is a function whose degree is 2 and where the *y* is not always exactly equal to the function. These types of functions use symbols called inequality symbols that include the symbols we know as less than, greater than, less than or equal to, and greater than or equal to. So, instead of seeing an equals sign, you will see these inequality symbols.

All quadratic inequalities are of the form *ax*^2 + *bx* + *c*, where *a*, *b*, and *c* are numbers. The numbers *b* and *c* can be 0, but *a* must equal a number. It cannot be 0. This is because our quadratic inequality must have an *x*^2 value. The other two terms do not need to be there.

For this lesson, we will work with the function *f(x)* > *x*^2 + 4*x* + 3. This function is a quadratic inequality because its degree is 2, meaning that its highest exponent is 2, and then it uses an inequality symbol instead of an equals sign.

The first step in graphing a quadratic inequality is to plot some points. The easiest way to do this is to find the high or low point of the graph. All quadratics graph into a parabola of some sort with a high or low point.

The *x* value of this point is calculated using this formula: *h* = -*b*/2*a*, where the *h* stands for the *x* value we are looking for. The *a* and the *b* stand for the numbers used in our function in their respective locations. Our *a* in our function is the number 1, and our *b* the number 4. Plugging these numbers into the formula, we get *h* = -4/2*1 = -4/2 = -2 for our *x* value.

Plugging this *x* value into our function, we can find our *y* value for that point. Doing this we get *y* = (-2)^2 + 4*(-2) + 3 = 4 - 8 + 3 = -1. So, that gives us our first point: (-2, -1). To find our other points, I'm going to choose a couple *x* values to the left and to the right of this point. I'm going to fill a table with my points before plotting them. Since my *x* for the high or low point is -2, the other numbers I am going to pick are -4, -3, -1, and 0. I have two points to the left and two points to the right. Filling in my table, I get this:

x |
y |
---|---|

-4 | 3 |

-3 | 0 |

-2 | -1 |

-1 | 0 |

0 | 3 |

Now I can go ahead and plot these points on my graph.

Now that we have all the points on the graph, we draw our line. But wait, we can't just draw any old line! The kind of line we draw depends on the symbol we have. The kind of line we have actually tells us whether the numbers on the line are included in the solution or not. If our symbol is greater than or less than, then we have a dashed line to tell us that the numbers that make up the line are not included in the solution.

But, if we have the greater than or equal to or the less than or equal to symbol, then the numbers that make up the line are included, and so we draw a solid line. So, what type of symbol do we have in our case? We have a greater than, so that means our line is dashed.

We're not done just yet! I know, normally we would be. But we are dealing with an inequality, after all! Notice how our quadratic inequality tells us that the solution includes all values greater than the function? So, to finish graphing and solving our quadratic, we need to figure out where to shade our graph to tell us what our solutions are.

Do we shade above or below the line? What exactly does the greater than symbol tell us? This symbol is telling us that the solutions are all those values of *y* that are bigger than the *y* values of our line. Does this answer our question of where to shade? Yes, it does! It tells us to shade above the line since it is those numbers that have a *y* value greater than our line.

We can test this by picking a random point above the line. Let's, say, compare the point (0, 5) with the point (0, 3). Which point has a bigger *y* value? Why, it's the (0, 5) point, which is above the line. So, we shade that part of our graph, and we are done! All of our solutions are shown on this graph now.

Now that we are finished, let's review what we've learned. We've learned that a **quadratic inequality** is a function whose degree is 2 and where the *y* is not always exactly equal to the function. Instead of an equals sign for our function, we have inequality symbols. These symbols include the less than symbol, the greater than symbol, the less than or equal to symbol, and the greater than or equal to symbol.

We know that all quadratics graph into a parabola of some sort. The difference between a graph of a quadratic inequality and that of a regular quadratic is that our graph has shading and our line may be different. We plot the points the same as we would a regular quadratic. But our line will be dashed if our symbols are either the less than or the greater than symbol; otherwise, the line is solid.

If our symbol is either the greater than symbol or the greater than or equal to symbol, we shade the part of the graph that is above the line. If our symbol is either the less than symbol or the less than or equal to symbol, we shade the part of the graph that is below the line.

Analyze the information in this lesson in order to:

- Define quadratic inequality
- Plot points on a graph
- Explain how to graph the line
- Remember where to shade a graph

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Math 105: Precalculus Algebra14 chapters | 124 lessons | 12 flashcard sets

- What is a Parabola? 4:36
- Parabolas in Standard, Intercept, and Vertex Form 6:15
- Identifying Conic Sections: General Form & Standard Form 6:22
- How to Factor Quadratic Equations: FOIL in Reverse 8:50
- Factoring Quadratic Equations: Polynomial Problems with a Non-1 Leading Coefficient 7:35
- How to Complete the Square 8:43
- Completing the Square Practice Problems 7:31
- How to Solve a Quadratic Equation by Factoring 7:53
- How to Use the Quadratic Formula to Solve a Quadratic Equation 9:20
- Graphing & Solving Quadratic Inequalities: Examples & Process 6:14
- Go to Quadratic Functions

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