Week 2, Day 2

Today we learned how to solve and graph systems of inequalities. The graph of the equation will normally separate the plane into two or more regions. In each region, one of the following must be true: a) All points in the region are solutions of the inequality. b) No point in the region is a solution of the inequality. Thus, you can determine whether the points in an entire region satisfy the inequality by simply testing one point in the region. The following are steps to graph an inequality in 2 variables:

1. Replace the inequality sign with an equal sign, and sketch the graph of the resulting equation. 

(Use a dashed line for < or > and a solid line for <(=) or >(=)). 

2. Test one point in each of the regions formed by the graph in Step 1. If the point satisfies the inequality, shade the entire region todenote that every point in the region satisfies the inequality.

3. A solution of a system of inequalities in x and y is a point (x , y) that satisfies each inequality.

4. For a system of inequalities, it is helpful to find the vertices of the solution region.

Here are some examples:

Example 1:

Sketch the graph of y >(=) x^2 - 1

The graph, as shown above, of the corresponding equation y = x^2 - 1 is a parabola. By testing the point (0,0) above the parabola and the point (0,-2) below the parabola, you can see that the points that satisfy the inequality are those lying above (or on) the parabola. This inequality is also nonlinear.

Example 2:

Sketch the graphs of x >(=) -2 and y <(=) 3.

a) The graph of the corresponding equation x = -2 is a vertical line. The points that satisfy the inequality are those lying to the right of this line. b) The graph of the corresponding equation y = 3 is a horizontal line. The points that satisfy the inequality are those lying below (or on) this line.

Example 3:

Sketch the region containing all points that satisfy the system. 

x^2 - y <(=) 1

-x + y <(=) 1

As shown in the graph above, the points that satisfy the inequality x^2 - y <(=) 1 are the points lying above (or on) the parabola given by y = x^2 - 1. The points that satisfy the inequality -x + y <(=) are rhe points lying below (or on) the line given by y = x + 1. To find the points of the intersection of the parabola and line, solve the system of equations. Using the method of substitution, you will find the solutions to be (-1,0) and (2,3).