Name: ________________
Algebra 1
WORKSHEET: COMPOUND INEQUALITIES
10/17/12
Compound Inequalities
Consider this statement: “Erik’s score on his algebra test was between 85 and 95.”
How could we write this using inequalities? And, how would we show the solution set using a
number line? If we look carefully at this statement, we can break it down into two parts: Erik’s score
is greater than 85, and Erik’s score is also less than 95 – we need two inequalities to express this
statement algebraically, and we need to indicate that both inequalities are true. Letting 𝑠 represent
Erik’s test score, we can write the first part using the basic inequality 𝑠 > 85, and the second part
using 𝑠 < 95. To represent the entire statement, we write both of these inequalities together, linked
by and.
𝑠 > 85 𝑎𝑛𝑑 𝑠 < 95
In Mathematics, as in English, when two statements are combined with the word and or the word or,
the result is called a compound sentence. The truth of a compound sentence can be determined if
the truth or falsity of the individual sentences can be determined. When two inequalities are linked
together using either and or or, we call the statement a compound inequality.
Compound Inequalities Using “AND”
When two inequalities are linked by and, we call the statement a conjunction, and the statement is
true only if both inequalities are true. The solution set of a conjunction is the set of values that are
solutions to both of the inequalities that make up the conjunction. In the example above, the solution
to the statement “Erik’s score in his algebra test was between 85 and 95” is the set of numbers that
are both greater than 85 and less than 95. On a number line, we can see that this is represented by
the part of the number line that is the overlap of the two separate inequalities 𝑠 > 85 and 𝑠 < 95.
The number lines below show this: the top two number lines represent the individual inequalities 𝑠 >
85 and 𝑠 < 95, and the bottom one shows the area of the overlap – this represents the solution set
of the compound inequality.
𝑠 > 85
𝑠 < 95
𝑠 > 85 𝑎𝑛𝑑 𝑠 < 95
►A compound inequality using and can also be written in a more compact way, without using the
“and:” 85 < 𝑠 < 95. Any compound inequality written in this form is always an ”and” type of
compound inequality. Graph the following conjunctions (”and” inequalities) on the number line:
1. 𝑥 > −3 𝑎𝑛𝑑 𝑥 < 5
2. 0 < 𝑦 < 7
3. 𝑧 > 5 𝑎𝑛𝑑 𝑧 < 3
Did you carefully graph the solution set for #3? If so, you found something interesting – there was
no overlap of the two inequalities 𝑧 > 5 and 𝑧 < 3. Since there is no value of 𝑧 that makes both of
these inequalities true, there is no solution to this compound inequality. Of course, when you think
about it, this makes sense – there is no number that is both less than 3 and greater than 5.
Compound Inequalities Using “OR”
When two sentences are joined by the word or, the resulting compound sentence is called a
disjunction. The inequality 𝑥 ≥ 3 is a form of disjunction because it can be written as 𝑥 > 3 𝑜𝑟 𝑥 =
3. A disjunction is true only if at least one of the statements is true. Thus the inequality 𝑥 ≥ 3 is true
if either 𝑥 > 3 is true or if 𝑥 = 3 is true. When two inequalities are linked with or to form a
disjunction, the solution set is the combination of the solution sets of the individual inequalities. For
example, let’s look at the disjunction 𝑦 < 0 𝑜𝑟 𝑦 > 5. As shown on the number lines below, the
solution set to this compound inequality is found by putting together the solution sets for 𝑦 < 0 and
𝑦 > 5:
𝑦<0
𝑦>5
𝑦 < 0 𝑜𝑟 𝑦 > 5
All of the values that satisfy 𝑦 < 0 are part of the solution set for 𝑦 < 0 𝑜𝑟 𝑦 > 5.
In addition, all of the values that satisfy 𝑦 > 5 are also part of the solution set for this disjunction. As
you can see, there are two distinct parts of the solution – separated by an “empty” region. This is
often the case when we graph solution sets of disjunctions.
Here are some disjunctions to graph in class:
4. 𝑎 < −3 𝑜𝑟 𝑎 > 2
5. 𝑏 ≤ 15 𝑜𝑟 𝑏 ≥ 25
6. 𝑐 < 7 𝑜𝑟 𝑐 > 4
What did you notice when you graphed this last disjunction (#6)? The two inequalities overlap – so
the solution set to the combination covers the entire number line – the solution set is “all real
numbers.” If you think for a minute about what these two inequalities say, when taken together with
or between them, you will see that this result makes sense – every number is either less than 7 or
greater than 4!
On the following pages are some homework problems. The first set consists of some compound
inequalities – graph them carefully, being sure to note whether each is a conjunction (and) or
disjunction (or). The second part consists of some simple inequalities (meaning not compound) that
will take some work to solve – don’t forget to distribute and combine like terms, just as you did in
solving similar types of equations. There will be problems like these on Friday’s test!
For al the problems that follow, write the solution using interval notation as well.
Graph the solutions to these compound inequalities and write using interval notation:
1. 𝑥 > 4 𝑎𝑛𝑑 𝑥 < 10
2. 𝑦 ≥ −7 𝑎𝑛𝑑 𝑦 < 0
3. 3 ≤ 𝑥 ≤ 12
4. 75 ≤ 𝑦 < 120
5. 2𝑥 + 5 < 9 𝑜𝑟 3𝑥 > 18
6. −2𝑦 > 8 𝑜𝑟 − 3𝑦 ≤ −21
7. 𝑧 ≤ 3 𝑜𝑟 𝑧 > 0
8. 𝑛 > 14 𝑎𝑛𝑑 𝑛 < 10
Solve these “simple” inequalities and write the solutions using interval notation – you do not need to
graph them!
9. 7𝑧 − 3 < 2𝑧 + 7
10. 𝑦 + 4 ≥ 3𝑦 + 12
11. 4(𝑥 + 6) ≥ 16
12. – (𝑥 + 7) ≤ −3
13. −3(𝑡 − 5) > 12
14. 6𝑥 + 7 < 4 + 2(3𝑥 − 2)
15. 7 + 5(2𝑥 − 3) ≤ 12
16. −2(𝑥 + 5) ≥ −2𝑥 − 18
17. 3 − 2(2𝑥 + 7) < 5
18. 6𝑥 − (𝑥 − 2) ≥ 3𝑥 + 8
19. 7𝑥 − 3(2𝑥 + 1) > 𝑥 − 3
20. 7𝑥 − 3(2𝑥 + 1) ≤ 𝑥 − 3