Compound Inequalities”
Tool Box:
Equivalent Inequalities
Properties of Inequalities
Summary:
Question:
Compound Inequality: consists of two separate inequalities joined by
“and” or “or”
“AND”: A compound inequality containing and is true only if BOTH inequalities are TRUE
The graph is the “intersection” of the graphs. The solution(s) must be the solution for both
inequalities
Graph an Intersection
EXAMPLE 1:
𝑥 < 3 𝑎𝑛𝑑 𝑥 ≥ −2
1. Graph 𝑥 < 3
Note that the graph does not include 3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
2. Graph 𝑥 ≥ −2
Note that the graph does include -2
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
3. Find the intersection of the two graphs
{𝑥 | − 𝟐 ≤ 𝑥 < 𝟑}
Note that you write the inequality values as
they appear in order on the number line
Solve and Graph an Intersection
Example 2:
−5 < 𝑥 − 4 < 2
Method # 1 Express the inequality using “and”
−5 < 𝑥 − 4
𝒂𝒏𝒅
𝑥 −4 < 2
−5 < 𝑥 − 4
𝒂𝒏𝒅
𝑥 −4 < 2
+4
+ 4
+4
+4
−1 < 𝑥
𝑥
< 6
The solution set is the intersection of the two
graphs
Solve, then graph the compound inequality
1. Write the inequalities separately
2. Train tracks and isolate the variable
3. Addition Property of Inequality (add 4 to
both sides of the inequality
4. Combine Like Terms/Simplify
{𝑥 | − 𝟏 < 𝑥 < 𝟔}
Graph
−1 < 𝑥
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Graph
𝑥 < 6
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8
Find the intersection
-6 -5 -4 - -2 -1
0 1 2 3 4 5 6 7 8
𝑜𝑟
𝑥 > −1
Method # 2 Do not separate the inequalities
−𝟓 < 𝒙 − 𝟒 < 𝟐
+4
+ 4
−1 <
𝑥
+4
<6
15 < −7𝑥 + 1 < 50
Example 3:
15 < −7𝑥 + 1 < 50
15
−1
14
14
−7
−2
-10
-10
-9
-9
-9
-8
-8
-8
-7
< −7𝑥 + 1 < 50
− 1
−1
< −7𝑥
< 49
−7𝑥
49
<
<
−7
−7
>
𝑥
> −7
-7
-7
-6
-6
-6
-5
-5
-5
-4
-4
-4
-3
-3
-3
-2
-2
-2
Solve and graph the compound inequality
1. Write the inequality
2. Train tracks/ Isolate the variable
3. Subtraction Property of Inequality
(Subtract 1 from each expression)
4. Combine Like Terms/Simplify
5. Division Property Inequality (Divide
each expression by -7)
6. Since you divided the inequality by a
negative number, FLIP both of the
inequality symbols and simplify
Solution Set
{𝑥 | − 7 < 𝑥 < −2}
-10
1 Write the compound inequality
2. Train tracks/Isolate the variable
3. Addition Property of Inequality (add 4 to
each expression)
4. Simplify
Graph
𝑥 > −7
Graph
𝑥 < −2
-1
-1
-1
Find the intersection:
{𝑥 | − 7 < 𝑥 < −2}
“OR”: A compound inequality containing “or” is
true only if one or more of the inequalities are
TRUE. The graph is the “union” of the graphs.
The solution is a solution of either inequality, not
necessarily both
Solve and Graph a Union
Example 4: −3ℎ + 4 < 19 𝑜𝑟 7ℎ − 3 > 18
−3ℎ + 4 < 19
−3ℎ + 4 < 19
− 4
−4
−3ℎ
< 15
−3ℎ
15
< −3
−3
𝒉 > −𝟓
OR
OR
𝒐𝒓
7ℎ − 3 > 18
7ℎ − 3 > 18
+ 3
+3
7ℎ
> 21
7ℎ
21
> 7
7
𝒉 > 𝟑
1 Solve the inequalities individually
2. Train Tracks/Isolate the variable
3. Subtraction Property (Subtract 4)
Addition Property (Add 3)
4. Combine Like Terms/Simplify
5. Isolate the variables/Division Property
of Inequality (Divide by -3, flip the
inequality, Divide by 7)
6. Simplify
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
8 9
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
8 9
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
8 9
Graph
ℎ > −5
Graph
ℎ > 3
Find the union
Notice that the graph of ℎ > −5 contains every point as the graph of ℎ > 3
So, the union is the graph of ℎ > −5 because it contains all the points that ℎ > 3 does
Solution Set / Set Notation
{𝒉| 𝒉 > −𝟓 }
Example 5: 5𝑥 + 6 ≤ −9 𝑜𝑟 2𝑥 − 8 > 12
5x + 6 ≤ −9
or
5x + 6 ≤ −9
− 6
−6
5𝑥
≤ −15
5𝑥
≤ −15
𝟓𝒙
−𝟏𝟓
≤
𝟓
𝟓
or
2x − 8 > 12
2x − 8 > 12
+8 +8
2𝑥
> 20
2𝑥
> 20
𝟐𝒙
𝟐𝟎
>
𝟐
𝟐
𝑥 ≤ −3
𝑥
> 10
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
8 9 10 11 12
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
8 9 10 11 12
-6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 7
𝑥 ≤ −3
𝑜𝑟
8 9 10 11 12
𝑥
> 10
Try It!!!!
1. −3 ≤ −2𝑥 + 1 < 11
2. 9𝑥 + 1 < −17 𝒐𝒓 7𝑥 − 12 > 9
1. Solve the inequalities separately
2. Train Tracks/ Isolate the variable
3. Subtraction Property of Inequality
(Subtract 6) / Addition Property (add 8)
4. Combine Like Terms/ Simplify
5. Isolate the variables
6. Division Property of Inequality
(Divide by 5 / Divide by 2)
7. Combine Like Terms /Simplify
Graph
𝑥 ≤ −3
Graph
𝑥
> 10
Find the union. Remember the solution of a
compound inequality using “or is a
solution of either inequality, not
necessarily both, so both graphs are
solutions
Solution Set
Solve and Graph the inequalities