Graphing & Solving Quadratic Inequalities: Examples & Process

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  • 1:09 Plotting Points
  • 2:58 The Line
  • 3:49 Shading
  • 5:01 Lesson Summary
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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

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 + 4x + 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.

Plotting the Points

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/2a, 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.

The points
graphing inequality

The Line

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.

The line
graphing inequality

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