Graph the solution set of the following system of inequalities

First, rewrite the two inequalities in slope-intercept form: y=mx+b.

For the first inequality, y-x-3≤0, you can add x and 3 to both sides to get y=x+3.

For the second inequality, 2x+3y>-6, add 2x to both sides and then divide by 3. That will give you y>-2/3x-2,.

Now graph the two inequalities: y≤x+3 and y>-2/3x-2. Since y≤x+3 is not a strict inequality (it has equal to as part of its sign), graph this inequality by drawing a solid line. Any area of the graph under or touching the line is part of the inequality. Graph y>-2/3x-2 with a dashed line, because it is a strict inequality (doesn't contain 'equal to' in the sign). Any area of the graph that is over this line is part of that inequality. Find the area of the graph the two inequalities overlap and shade in the region below y≤x+3 and above y>-2/3x-2. The shaded area is the solution to this problem.

First, take a very simple inequality, y

This kind of region is called a half-plane because it is one of two parts of the plane into which a boundary line divides it. In this case, the region consists of all those points that lie on and above the line y = 1.

Another example is y

It is also a half-plane. In this case, the solution set consists of all points in the half-plane including and below the line y = 1.

In both cases, the equation of the boundary line is found by replacing the inequality symbol with an equals sign.

Consider y

The solution sets for both inequalities are shown below.

The following general key idea is always true.

Key Idea

For any linear inequality, if the inequality symbol is replaced with an equals sign, the result is a line that divides the plane into two half-planes. The solution set for the inequality is one of these half-planes.

Example 1

Graph y

Solution

First, replace the inequality symbol in y 2x - 3 with an equals sign, in this case y = 2x - 3. Graph the line.

Now, since the inequality states that the y-coordinate is greater than or equal to the linear expression in x , the solution set for the inequality is the set of points above this line.

This is shown in the shaded region above.

If the inequality symbol were reversed and the inequality was y

Key Idea

• If a linear inequality sets y greater than or equal to the linear expression in x, then the solution set is the set of points above the boundary line.

• If a linear inequality sets y less than or equal to the linear expression in x, then the solution set is the set of points below the boundary line.

Using this key idea, the solution set for y

So far, only inequalities containing

Key Idea

• When the inequality symbol is

• When the inequality symbol is < or >, draw a dashed line on the boundary of the half-plane to indicate that the boundary line is not included.

This key idea is illustrated by the graphs shown below.

How do you graph the solution set of a system of inequalities?

What is the solution of the system of inequalities system?