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Graphing Compound Inequalities

Graphing the Solution Set to a Compound Inequality

The soloution set to a compound inequality such as

x > 2 and x < 5

consists of all numbers that are in the solution sets to both simple inequalities. So the solution set to this compound inequality is the intersection of those two solution sets. In symbols,

{x | x > 2 and x < 5} = {x | x > 2} Ç {x | x < 5}.

 

Example 1

Graphing compound inequalities

Graph the solution set to the compound inequality x > 2 and x < 5.

Solution

We first sketch the graph of x > 2 and then the graph of x < 5, as shown in the top two number lines in the figure below. The intersection of these two solution sets is the portion of the number line that is shaded on both graphs, just the part between 2 and 5, not including endpoints. The graph of {x | x > 2 and x < 5} is shown at the bottom of the figure. We write this set in interval notation as (2, 5).

The solution set to a compound inequality such as

x > 4 or x < -1

consists of all numbers that satisfy one or the other or both of the simple inequalities. So the solution set to the compound inequality is the union of the solution sets to the simple inequalities. In symbols,

{x | x > 4 or x < -1} = {x | x > 4 } È {x | x < -1}

 

Example 2

Graphing compound inequalities

Graph the solution set to the compound inequality x > 4 or x < -1.

Solution

To find the union of the solution sets to the simple inequalities, we sketch their graph as shown at top of the figure that appears below. We graph the union of these two sets by putting both shaded regions together on the same line as shown in the bottom graph of the figure. This set is written in interval notation as

(-∞, -1) È (4, )

 

Caution

When graphing the intersection of two simple inequalities, do not draw too much. For the intersection, graph only numbers that satisfy both inequalities. Omit numbers that satisfy one but not the other inequality. Graphing a union is usually easier because we can simply draw both solution sets on the same number line.

It is not always necessary to graph the solution set to each simple inequality before graphing the solution set to the compound inequality.We can save time and work if we learn to think of the two preliminary graphs but draw only the final one.

 

Example 3

Overlapping intervals

Sketch the graph and write the solution set in interval notation to each compound inequality.

a) x < 3 and x < 5

b) x > 4 or x > 0

Solution

a) To graph x < 3 and x < 5, we shade only the numbers that are both less than 3 and less than 5. So numbers between 3 and 5 are not shaded in the figure below. The compound inequality x < 3 and x < 5 is equivalent to the simple inequality x < 3. The solution set can be written as (-∞, 3).

b) To graph x > 4 or x > 0, we shade both regions on the same number line as shown on the figure below. the compound inequality x > 4 or x > 0 is equivalent to the simple inequality x > 0. The solution set is (0, )

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