1. Solving Compound Inequalities
Section 8.4
Solving Compound Inequalities

2. Objectives Find the intersection and the union of two sets
Solve compound inequalities containing the word and
Solve double linear inequalities
Solve compound inequalities containing the word or

3. Objective 1: Find the Intersection and the Union of Two Sets
When the word and or the word or is used to connect pairs of inequalities, we call the statement a compound inequality. To solve compound inequalities, we need to know how to find the intersection and union of two sets.
The Intersection of Two Sets The intersection of set A and set B, written A ∩ B, is the set of all elements that are common to set A and set B.
The Union of Two Sets The union of set A and set B, written A U B, is the set of elements that belong to set A or set B, or both.

4. Objective 1: Find the Intersection and the Union of Two Sets
Venn diagrams can be used to illustrate the intersection and union of sets. The area shown in purple in figure (a) represents A ∩ B and the area shown in both shades of red in figure (b) represents A U B.
Read as “A intersect B.”
Read as “A union B.”

5. EXAMPLE 1
Let A = {0, 1, 2, 3, 4, 5, 6} and B = {–4, –2, 0, 2, 4}
a. Find A ∩ B b. Find A U B
Strategy In part (a), we will find the elements that sets A and B have in common, and in part (b), we will find the elements that are in one set or the other set, or both.
Why The symbol ∩ means intersection, and the symbol U means union.

6. EXAMPLE 1
Let A = {0, 1, 2, 3, 4, 5, 6} and B = {–4, –2, 0, 2, 4}
a. Find A ∩ B b. Find A U B
Solution
a. Since the numbers 0, 2, and 4 are common to both sets A and B, we have
A ∩ B = {0, 2, 4}
Since the intersection is, itself, a set, braces are used.
b. Since the numbers in either or both sets are –4, –2, 0, 1, 2, 3, 4, 5 and 6, we have
A U B = {−4, −2, 0, 1, 2, 3, 4, 5, 6}
0, 2, and 4 are not listed twice.

7. Objective 2: Solve Compound Inequalities Containing the Word And
When two inequalities are joined with the word and, we call the statement a compound inequality.
The solution set of a compound inequality containing the word and includes all numbers that make both of the inequalities true. That is, it is the intersection of their solution sets.
In the following figure, the graph of the solution set of x ≥ –3 is shown in red, and the graph of the solution set of x ≤ 6 is shown in blue.
Preliminary work to determine the graph of the solution set

8. Objective 2: Solve Compound Inequalities Containing the Word And
The solution set of x ≥ –3 and x ≤ 6 is the bounded interval [–3, 6], where the brackets indicate that the endpoints, –3 and 6, are included. It represents all real numbers between –3 and 6, including –3 and 6. Intervals such as this, which contain both endpoints, are called closed intervals.
Since the solution set of x ≥ –3 and x ≤ 6 is the intersection of the solution sets of the two inequalities, we can write

9. Objective 2: Solve Compound Inequalities Containing the Word And
The solution set of the compound inequality x ≥ –3 and
x ≤ 6 can be expressed in several ways:
As a graph:
In words: all real numbers between –3 to 6
In interval notation: [–3, 6]
Using set-builder notation: {x | x ≥ –3 and x ≤ 6}

10. Objective 2: Solve Compound Inequalities Containing the Word And
A bounded interval that includes only one endpoint, is called a half-open interval. The following chart shows the various types of bounded intervals, along with the inequalities and interval notation that describe them.

11. Solve > 0 and 2x – 3 < 5. Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 2
Strategy We will solve each inequality separately. Then we will graph the two solution sets on the same number line and determine their intersection.
Why The solution set of a compound inequality containing the word and is the intersection of the solution sets of the two inequalities.

12. Solve > 0 and 2x – 3 < 5. Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 2
Solution
In each case, we can use properties of inequality to isolate the variable on one side of the inequality.
Next, we graph the solutions of each inequality on the same number line and determine their intersection.
Preliminary work to determine the graph of the solution set

13. Solve > 0 and 2x – 3 < 5. Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 2
Solution
We see that the intersection of the graphs is the set of all real numbers between –2 and 4. The solution set of the compound inequality is the interval (–2, 4), whose graph is shown below. This bounded interval, which does not include either endpoint, is called an open interval. Written using set-builder notation, the solution set is {x | x > –2 and x < 4}.
The graph of the solution set

14. Objective 3: Solve Double Linear Inequalities
Inequalities that contain exactly two inequality symbols are called double inequalities. An example is
Double Linear Inequalities The compound inequality c < x < d is equivalent to c < x and x < d.

15. EXAMPLE 5
Solve –3 ≤ 2x + 5 < 7. Graph the solution set and write it using interval notation and set-builder notation.
Strategy We will solve the double inequality by applying properties of inequality to all three of its parts to isolate x in the middle.
Why This double inequality –3 ≤ 2x + 5 < 7 means that –3 ≤ 2x + 5 and 2x + 5 < 7. We can solve it more easily by leaving it in its original form.

16. EXAMPLE 5
Solve –3 ≤ 2x + 5 < 7. Graph the solution set and write it using interval notation and set-builder notation.
Solution
The solution set of the double linear inequality is the half-open interval [–4, 1), whose graph is shown below. Written using set-builder notation, the solution set is {x | –4 ≤ x < 1}.

17. Objective 4: Solve Compound Inequalities Containing the Word Or
When two inequalities are joined with the word or, we also call the statement a compound inequality.
The solution set of a compound inequality containing the word or includes all numbers that make one or the other or both inequalities true. That is, it is the union of their solution sets.
In the following figure, the graph of the solution set of x ≤ –3 is shown in red, and the graph of the solution set of x ≥ 2 is shown in blue.

18. Objective 4: Solve Compound Inequalities Containing the Word Or
For the compound inequality of x ≤ –3 or x ≥ 2, we can write the solution set as the union of two intervals:
We can express the solution set of the compound inequality x ≤ –3 or x ≥ 2 in several ways:
As a graph:
In words: all real numbers less than or equal to –3 or greater than or equal to 2
As the union of two intervals: (–∞, –3] U [2, ∞)
Using set-builder notation: {x | x ≤ –3 or x ≥ 2}

19. Solve
Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 6
Strategy We will solve each inequality separately. Then we will graph the two solution sets on the same number line to show their union.
Why The solution set of a compound inequality containing the word or is the union of the solution sets of the two inequalities.

20. EXAMPLE 6 Solution To solve each inequality, we proceed as follows:
Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 6
Solution
To solve each inequality, we proceed as follows:
Next, we graph the solutions of each inequality on the same number line and determine their union.
Preliminary work to determine the graph of the solution set

21. Solve
Graph the solution set and write it using interval notation and set-builder notation.
EXAMPLE 6
Solution
The union of the two solution sets consists of all real numbers less than –1 or greater than 2. The solution set of the compound inequality is the union of two intervals: (–∞, –1) U (2, ∞). Its graph appears below. Written using set-builder notation, the solution set is {x | x > 2 or x < –1}.
The graph of the solution set