1. Compound Inequalities
A compound inequality is either two inequalities separated by a word, or an expression in between two inequality symbols.

2. Compound Inequalities
A compound inequality is either two inequalities separated by a word, or an expression in between two inequality symbols.
Let’s look at the first type…separated by a word

3. Compound Inequalities
There are two words that can appear in between inequalities
AND / OR

4. Compound Inequalities
There are two words that can appear in between inequalities
AND / OR
With AND, the solution set is what is shared or common

5. Compound Inequalities
There are two words that can appear in between inequalities
AND / OR
With AND, the solution set is what is shared or common
With OR, the solution set is all solutions combined

6. Compound Inequalities
There are two words that can appear in between inequalities
AND / OR
With AND, the solution set is what is shared or common
With OR, the solution set is all solutions combined
We are going to use a line graph to help us find the solution set…

7. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15

8. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality

9. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
2𝑥+5>−15

10. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
+2=+2
3𝑥≤12
3𝑥 3 ≤ 12 3
𝑥≤4
2𝑥+5>−15
−5=−5
2𝑥>−20
2𝑥 2 > −20 2
𝑥>−10

11. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points

12. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
- start with 𝑥≤4, it’s a closed circle with the arrow going left
-10 4

13. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
- start with 𝑥≤4, it’s a closed circle with the arrow going left
- now graph 𝑥>−10. open circle with arrow pointing right
-10 4

14. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
STEP # 3 : With AND in between the expressions, we need to find where the
graphs are ON TOP of each other
-10 4

15. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
STEP # 3 : With AND in between the expressions, we need to find where the
graphs are ON TOP of each other
THIS IS OUR SHARED AREA
-10 4

16. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
STEP # 3 : With AND in between the expressions, we need to find where the
graphs are ON TOP of each other
Mesh the ”shared area “ into one line
-10 4

17. Compound Inequalities
EXAMPLE #1 : Find the solution for
3𝑥−2≤10 𝐴𝑁𝐷 2𝑥+5>−15
STEP # 1 : Solve each inequality
3𝑥−2≤10
𝑥≤4
2𝑥+5>−15
𝑥>−10
STEP # 2 : Set up a number line and graph your points
STEP # 3 : With AND in between the expressions, we need to find where the
graphs are ON TOP of each other
Mesh the ”shared area “ into one line
-10 4
Solution is −10<𝑥≤4

18. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11

19. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11
STEP #1 : Solve each inequality
6−𝑥≤1
− =−6
−𝑥≤−5
−𝑥 −1 ≤ −5 −1
𝑥≥5
4𝑥−1>11
+1=+1
4𝑥>12
4𝑥 4 > 12 4
𝑥>3
Remember, when you divide by a negative, the inequality changes direction

20. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11
STEP #1 : Solve each inequality
6−𝑥≤1
𝑥≥5
4𝑥−1>11
𝑥>3
STEP #2 : Set up a number line, graph your points and draw your arrows 3 5

21. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11
STEP #1 : Solve each inequality
6−𝑥≤1
𝑥≥5
4𝑥−1>11
𝑥>−5
closed circle and to the right
open circle and to the right
STEP #2 : Set up a number line, graph your points and draw your arrows 3 5

22. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11
STEP #1 : Solve each inequality
6−𝑥≤1
𝑥≥5
4𝑥−1>11
𝑥>−5
closed circle and to the right
open circle and to the right
STEP #2 : Set up a number line, graph your points and draw your arrows
FINAL graph 3 5
STEP # 3 : With OR in between the expressions, we will look at all possible solutions. Since the arrow for (3) joins and is in the same direction as (+5), the solution set is all numbers greater than (3)

23. Compound Inequalities
EXAMPLE #2 : Find the solution for
6−𝑥≤1 𝑂𝑅 4𝑥−1>11
STEP #1 : Solve each inequality
6−𝑥≤1
𝑥≥5
4𝑥−1>11
𝑥>−5
closed circle and to the right
open circle and to the right
STEP #2 : Set up a number line, graph your points and draw your arrows
FINAL graph 3 5
Solution is 𝑥>3

24. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.

25. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13

26. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = +8
3𝑥<21
If I covered the left side of the equation and only solved for the right side, I would add 8 to both sides…

27. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = = +8
9<3𝑥<21
When solving these types, whatever you would do to solve one side, you would also do to solve the other side…

28. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = = +8
9<3𝑥<21
9 3 < 3𝑥 3 < 21 3
Now divide EVERYTHING by the coefficient of 𝑥 which is 3

29. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = = +8
9<3𝑥<21
9 3 < 3𝑥 3 < 21 3
3<𝑥<7

30. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = = +8
9<3𝑥<21 3 7
9 3 < 3𝑥 3 < 21 3
Set up your number line and graph the points…
3<𝑥<7
Open circles

31. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
Solution is 3<𝑥<7
+8 = = +8
9<3𝑥<21 3 7
9 3 < 3𝑥 3 < 21 3
If both inequality symbols point to the LEFT, the solution is SQUEEZED in between your points…
3<𝑥<7
Open circles

32. Compound Inequalities
The other type of compound inequality that occurs is when there is an expression squeezed in between two inequalities.
EXAMPLE # 3 : <3𝑥−8<13
+8 = = +8
9<3𝑥<21 3 7
9 3 < 3𝑥 3 < 21 3
If both inequality symbols point to the LEFT, the solution is SQUEEZED in between your points…
3<𝑥<7
Open circles 3 7
If both inequality symbols point to the RIGHT, the solution HAS A GAP in between your points…
Solution is 3>𝑥>7

33. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3

34. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3
With this in mind, we solve absolute value equations like this…
EXAMPLE : Find the solution set for 𝑥+3 <10

35. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3
With this in mind, we solve absolute value equations like this…
EXAMPLE : Find the solution set for 𝑥+3 <10
−10<𝑥+3<10
Set up a compound inequality with the negative of the original answer on the left side. Keep the inequality symbol pointed in the same direction. Solve like we did with compound inequalities.

36. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3
With this in mind, we solve absolute value equations like this…
EXAMPLE : Find the solution set for 𝑥+3 <10
−10<𝑥+3<10
−3 = −3=−3
Subtract 3

37. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3
With this in mind, we solve absolute value equations like this…
EXAMPLE : Find the solution set for 𝑥+3 <10
−10<𝑥+3<10
−3 = −3=−3
−13<𝑥<7
Subtract 3

38. Compound Inequalities
ABSOLUTE VALUE INEQUALITIES
Recall that all absolute values are positive.
So the equation 𝑥 =3 has two solutions… 𝑥=−3 , 3
With this in mind, we solve absolute value equations like this…
EXAMPLE : Find the solution set for 𝑥+3 <10
−10<𝑥+3<10
−3 = −3=−3
−13<𝑥<7
** If we would graph this, both inequalities point left so a “squeeze”
Subtract 3
−13 7