Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
A Quadratic Inequality
First, what is a quadratic inequality? Well, the word quadratic tells us that we will have an x^2 term and that 2 will be our highest exponent. We might have an x term as well as a constant term, but these aren't required for quadratics. The inequality tells us that we are dealing with a function that may not be exactly equal. So we are dealing with greater than, less than, etc.
So we can define a quadratic inequality as a function of a quadratic that may not be exactly equal. We know that a quadratic equation looks like y = ax^2 + bx + c. Our quadratic inequality looks very similar except that we don't have an equal sign. Instead of an equal sign, we will have greater than, less than, greater than or equal to, or a less than or equal to symbol.
I've written these in standard form, where a, b, and c stand for numbers. But remember that for a quadratic, the only term that must be there is the first x^2 term. What does this mean? It means that while b and c can be 0, essentially deleting those terms, a must be a number and cannot be 0. If we plug in random numbers, we can see how a quadratic inequality will look.
We see that we can have one term, or two, or all three. But the one term that has to stay is the x^2 term.
System of Quadratic Inequalities
While a quadratic inequality is just one of those functions, a system of quadratic inequalities is a collection of quadratic inequality functions considered as a set. What this means is that we will have more than one quadratic inequality, and when we consider these inequalities, we consider them as a whole. Any solution to one function must also be a solution to the other functions in the system.
When you have only one function, a solution only needs to satisfy that one function, but for a system, the solution must satisfy all the functions in the system. In math, we have a special notation for systems. We use a big curly bracket in front of the system like this:
Graphing a Quadratic Inequality
A good way to see the solutions to a system of quadratic inequalities is to graph it. Graphing a system of quadratic inequalities is very similar to graphing a quadratic function by itself. The only difference is that we will be shading the area where the solutions are, and also the line we draw is determined by the inequality sign. Let's discuss these things by first graphing our first quadratic inequality: y > x^2.
As with other quadratics, we can either punch this function into a graphing calculator or make a table with points to plot. What I'm going to do is make a table with points. I'm going to choose the x values and calculate the y values using the x values. The x values I'm going to choose are -2, -1, 0, 1, and 2. To find each point, I plug in my x value wherever I see an x and then calculate to find my y. So, the first point is y = (-2)^2 = 4, which means my point is (-2, 4). I do this for each point to get a complete table.
I plot these points on a graph to get my curve.
To draw my curve, I need to look at the inequality sign I have. I see that I have a greater than symbol, so that means my line will be dashed. When graphing inequalities, if my symbol is greater than or less than, then my line is dashed. If the symbol is greater than or equal to or less than or equal to, then my line will be solid. I draw a dashed line for my curve since my symbol is greater than.
Also, because we are dealing with inequalities, we are going to shade a portion of the graph. Do I shade above or below the line? Let's think about it. The sign is greater than, so that means I need the part whose y values are greater than the y values of my line. I pick a point on the line, any point to see. I pick the point (1, 1). I ask myself which direction I need to go to get bigger y values. It's up. I need to go up, so that means I need to shade the part of the graph that is above the line.
I'm done graphing this function. Now I can move on to the rest.
Graphing the System of Quadratic Inequalities
Because I am dealing with a system of quadratic inequalities, I will be graphing everything on the same graph. I'm going to make a table for the points for the second and third equation in my system.
For the second equation, y <= 2x^2 + 4, I have this table for the points.
My symbol is less than or equal to, so that means my line is solid. I am shading the part that is under the line because these are the y values that are less than my line.
I've drawn two of my inequalities on the graph so far, and I see that there is a part of the graph where my shading overlaps. This is the area where the solutions work for both of these inequalities. What will happen when I draw the third inequality on the graph? Let's see. I'm going to make a table for the points for my third and final inequality.
For the third quadratic inequality, y < -x^2 + 3x + 5, my table is this:
Because these points get me barely past the point where the graph curves, I'm going to add another point just to see how the curve curves.
My symbol is less than, so my line is dashed and I shade the part that is under my line.
The solution to my system is the area where all my shading overlaps. So it is the area that is above the red dashed line, below the solid blue line, and below the dashed green line. All my values inside this region are solutions to my system.
What have we learned? We've learned that a quadratic inequality is a function of a quadratic that may not be exactly equal. A system of quadratic inequalities is a collection of quadratic inequality functions considered as a set. To mathematically notate a system, we use a big curly bracket in front of the functions.
To graph my system, I plot points for each function separately and graph each on the same graph. I use a dashed line for inequalities that are greater than or less than and a solid line for inequalities that are greater than or equal to or less than or equal to. If my inequality is either greater than or greater than or equal to, I shade above the line. If the inequality is less than or less than or equal to, then I shade below the line. The solution to my system is the part of the graph where all of the shading overlaps.
Once you've finished this lesson you should be able to graph a quadratic inequality and a quadratic inequality system in order to find the system's solution set.
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