5.5 - Systems of Inequalities

Graph of an Inequality

When we were dealing with one variable, the graph of an inequality was some interval on the real number line. Now we are dealing with two variables, and the graph of an inequality is the set of all ordered pairs (x, y) that satisfy the inequality. Typically, this will be represented by a half-plane.

Sketching an Inequality in Two Variables

1. Replace the inequality with an equal sign and sketch the resulting equation. Use a solid line if the equal to is included (<= or >=) and a dashed line of the inequality is strict (< or >).
2. Pick a test point not on the graph of the equation. If that test point satisfies the inequality, then shade every point on the side of the inequality that includes your test point. If the test point doesn't satisfy the inequality, then shade the other side of the inequality.

Solving a System of Inequalities

The solution to a system of inequalities is the set of all ordered pairs (x, y) that satisfy all the inequalities simultaneously.

The solution to the system of linear inequalities given is sketched to the right.

( x - 3)² + ( y - 3)² < 4
x + y <= 5
2x + 3y <= 12

Notice the circle is dotted (dashed) instead of solid. This is because there is a strict inequality (< or >), that is, the equal to is not included.

Now, one misleading thing about generating the output on a computer is that it looks very easy to sketch the system. In reality, there are four ways you can sketch the solution to a system of linear inequalities

1. Draw all the lines and curves. Pick a test point in each region (there will be many regions - there are 8 in the figure shown). Drawback - way too much work.
2. Shade each inequality as it is encountered. Whatever is shaded for each inequality is the solution. If there are three inequalities, whatever is shaded three times is the solution. If there are five inequalities, then whatever is shaded five times is the solution. Drawback - if you do a good job of shading, and you should, it becomes difficult to tell what is shaded three times as opposed to what is shaded four times.
3. Shade each inequality as it is encountered, but shade the false part in each case. The contrapositive of "if doesn't work, then shaded" is "if not shaded, then works". In other words, whatever is left unshaded will be the part that is the solution. Drawback - your answers will be unshaded where the books will be shaded.
4. Shade just the correct edge of each inequality. Then you can extend those to see which regions are shaded on all sides.

Whichever technique you use, write the word "yes" or something similar in the solution region.

Determining the System from the Graph.

1. Write the equation for each boundary on the graph.
2. Pick a test point in the solution region and substitute it into the equation.
3. Replace the equal sign by whichever inequality is needed to make the statement true. Include the equal sign if it is a solid line and don't include the equal sign in the inequality if it is a dashed line.