what would the graph of this look like
xy<0
You'd recognise it. :-)
xy < 0 means xy is negative.
When x is 1, what values can y have? When x is 5? pi? 1,000,000,000?
When x is -1, what values can y have?
If the product of two numbers, say x * y, is negative, what do you know about those numbers? One is p------- and the other is n-------
So the graph of xy<0 is all the set of (x,y) pairs where x is p----- and y is n------ plus the set of (x,y) pairs where x is n----- and y is p------.
Note that neither x or y can be zero; that means the graph doesn't touch the axes.
The graph of the inequality xy < 0 represents all the points that satisfy the inequality. To understand how to graph it, we can break it down into two separate inequalities: x > 0 and y > 0, and x < 0 and y < 0.
First, let's consider the case where x > 0 and y > 0. This means that both x and y are positive. In this region, the product (xy) will always be positive, so it does not satisfy the inequality xy < 0. Therefore, we shade this region.
Next, let's consider the case where x < 0 and y < 0. This means that both x and y are negative. Similar to the previous case, the product (xy) will also be positive in this region. So, we shade this region as well.
The shaded regions represent the regions of the graph where the product xy is positive, not satisfying the inequality xy < 0.
The graph of xy < 0 will then be the complement of the shaded region, which is the regions outside of these shaded regions. It will consist of four regions: (1) all points where x > 0 and y < 0, (2) all points where x < 0 and y > 0, (3) the x-axis (where y = 0), and (4) the y-axis (where x = 0). These regions satisfy the inequality xy < 0.
The resulting graph will consist of two diagonal lines that intersect at the origin (0,0), dividing the coordinate plane into four quadrants. The shaded regions will be outside these lines, including both the x-axis and the y-axis.
It's worth noting that the graph of xy < 0 is symmetric about the origin, meaning that if a point (a,b) satisfies the inequality, its reflection (-a, -b) will also satisfy it.
I hope this explanation helps you visualize the graph of xy < 0! Let me know if you have any further questions.