1. Solving Inequalities Lesson 1-5 Part 2
Algebra 2
Solving Inequalities
Lesson 1-5 Part 2

2. Goals Goal Rubric To write and solve compound inequalities.
Level 1 – Know the goals.
Level 2 – Fully understand the goals.
Level 3 – Use the goals to solve simple problems.
Level 4 – Use the goals to solve more advanced problems.
Level 5 – Adapts and applies the goals to different and more complex problems.

3. Big Idea: Compound Inequalities
Essential Question
Big Idea: Compound Inequalities
In Part 1 of the lesson, you learned how to solve and graph inequalities.
In this lesson you will use what you learned to write and solve compound inequalities.

4. Vocabulary
Compound Inequality

5. Definition
The inequalities you have seen so far are simple inequalities. When two simple inequalities are combined into one statement by the words AND or OR, the result is called a compound inequality.
Compound Inequality – the result of combining two inequalities. The words and and or are used to describe how the two parts are related.

6. What is the Difference Between and and or?
AND means intersection
-what do the two items have in common?
OR means union
-if it is in one item, it is in the solution A B

7. Disjunction
A disjunction is a compound statement that uses the word or.
Disjunction: x ≤ –3 OR x > x ≤ –3 U x > 2
A disjunction is true if and only if at least one of its parts is true.

8. Number Line and Compound Inequalities
You can graph the solutions of a compound inequality involving OR by using the idea of combining regions. The combined regions are called the union and show the numbers that are solutions of either inequality.
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9. Conjunction
A conjunction is a compound statement that uses the word and.
Conjunction: x ≥ –3 AND x < x ≥ –3 ⋂ x < 2
A conjunction is true if and only if all of its parts are true. Conjunctions can be written as a single statement as shown.
x ≥ –3 and x< –3 ≤ x < 2

10. Number Line and Compound Inequalities
You can graph the solutions of a compound inequality involving AND by using the idea of an overlapping region. The overlapping region is called the intersection and shows the numbers that are solutions of both inequalities.

11. Dis- means “apart.” Disjunctions have two separate pieces.
Con- means “together” Conjunctions represent one piece.
Reading Math
The “and” compound inequality y < –2 and y < 4 can be written as –2 < y < 4.
The “or” compound inequality y < 1 or y > 9 must be written with the word “or.”
Writing Math

12. Compound Inequalities

13. Example: Writing Compound Inequalities
Write the compound inequality shown by the graph.
The shaded portion of the graph is not between two values, so the compound inequality involves OR.
On the left, the graph shows an arrow pointing left, so use either < or ≤. The solid circle at –8 means –8 is a solution so use ≤.
x ≤ –8
On the right, the graph shows an arrow pointing right, so use either > or ≥. The empty circle at 0 means that 0 is not a solution, so use >.
x > 0
The compound inequality is x ≤ –8 OR x > 0.

14. Example: Writing Compound Inequalities
Write the compound inequality shown by the graph.
The shaded portion of the graph is between the values –2 and 5, so the compound inequality involves AND.
The shaded values are on the right of –2, so use > or ≥. The empty circle at –2 means –2 is not a solution, so use >.
m > –2
The shaded values are to the left of 5, so use < or ≤. The empty circle at 5 means that 5 is not a solution so use <.
m < 5
The compound inequality is m > –2 AND m < 5 (or –2 < m < 5).

15. Your Turn: Write the compound inequality shown by the graph. x > –9
The shaded portion of the graph is between the values –9 and –2, so the compound inequality involves AND.
The shaded values are on the right of –9, so use > or . The empty circle at –9 means –9 is not a solution, so use >.
x > –9
The shaded values are to the left of –2, so use < or ≤. The empty circle at –2 means that –2 is not a solution so use <.
x < –2
The compound inequality is –9 < x AND x < –2 (or –9 < x < –2).

16. Your Turn: Write the compound inequality shown by the graph. x ≤ –3
The shaded portion of the graph is not between two values, so the compound inequality involves OR.
On the left, the graph shows an arrow pointing left, so use either < or ≤. The solid circle at –3 means –3 is a solution, so use ≤.
x ≤ –3
On the right, the graph shows an arrow pointing right, so use either > or ≥. The solid circle at 2 means that 2 is a solution, so use ≥.
x ≥ 2
The compound inequality is x ≤ –3 OR x ≥ 2.

17. Example: Solve the compound inequality. Then graph the solution set.
6y < –24 OR y +5 ≥ 3
Solve both inequalities for y.
6y < –24
y + 5 ≥3 or
y < –4
y ≥ –2
The solution set is all points that satisfy y < –4 or y ≥ –2.
–6 –5 –4 –3 –2 –

18. Example: Solve the compound inequality. Then graph the solution set.
Solve both inequalities for c.
and
2c + 1 < 1
c ≥ –4
c < 0
The solution set is the set of points that satisfy both c ≥ –4 and c < 0 (-4 ≤ c < 0).
–6 –5 –4 –3 –2 –

19. Example: Solve the compound inequality. Then graph the solution set.
x – 5 < –2 OR –2x ≤ –10
Solve both inequalities for x.
x – 5 < –2 or –2x ≤ –10
x < x ≥ 5
The solution set is the set of all points that satisfy x < 3 or x ≥ 5.
–3 –2 –

20. Your Turn: Solve the compound inequality. Then graph the solution set.
x – 2 < 1 OR 5x ≥ 30
Solve both inequalities for x.
x – 2 < 1
5x ≥ 30 or
x ≥ 6
x < 3
The solution set is all points that satisfy x < 3 U x ≥ 6. –

21. Your Turn: Solve the compound inequality. Then graph the solution set.
2x ≥ –6 AND –x > –4
Solve both inequalities for x.
2x ≥ – and –x > –4
x ≥ – x < 4
The solution set is the set of points that satisfy both x ≥ –3 ⋂ x < 4.
–4 –3 –2 –

22. Your Turn: Solve the compound inequality. Then graph the solution set.
x –5 < 12 OR 6x ≤ 12
Solve both inequalities for x.
x –5 < or x ≤ 12
x < x ≤ 2
Because every point that satisfies x < 17 also satisfies x ≤ 2, the solution set is x < 17.

23. Your Turn: Solve the compound inequality. Then graph the solution set.
–3x < –12 AND x + 4 ≤ 12
Solve both inequalities for x.
–3x < – and x + 4 ≤ 12
x < – x ≤ 8
The solution set is the set of points that satisfy both 4 < x ≤ 8.

24. Example: Solve the compound inequality and graph the solutions.
Since 1 is added to x, subtract 1 from each part of the inequality.
–5 < x + 1 AND x + 1 < 2 –1
The solution set is
{x:–6 < x AND x < 1}.
–6 < x
x < 1
AND
-6 < x < 1
Graph -6 < x < 1. 1
–10 –8 –6 –4 –2 2 4 6 8 10

25. Example: Solve the compound inequality and graph the solutions.
Since 1 is subtracted from 3x, add 1 to each part of the inequality.
8 < 3x – 1 ≤ 11
9 < 3x ≤ 12
Since x is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication.
3 < x ≤ 4
The solution set is 3 < x ≤ 4. –5 –4 –3 –2 –1 1 2 3 4 5

26. Your Turn: Solve the compound inequality and graph the solutions.
Since 10 is subtracted from x, add 10 to each part of the inequality.
–9 < x – 10 < –5
–9 < x – 10 < –5
1 < x < 5
The solution set is {x:1 < x < 5}.
1 < x < 5
Graph 1 < x < 5. –5 –4 –3 –2 –1 1 2 3 4 5

27. Your Turn: Solve the compound inequality and graph the solutions.
Since 5 is added to 3n, subtract 5 from each part of the inequality.
–4 ≤ 3n + 5 < 11
– – 5 – 5
–9 ≤ 3n < 6
Since n is multiplied by 3, divide each part of the inequality by 3 to undo the multiplication.
–3 ≤ n < 2
The solution set is {n:–3 ≤ n < 2}.
Graph -3 ≤ x < 2. –5 –4 –3 –2 –1 1 2 3 4 5

28. Assignment
Section 1-5 part 2, Pg 40 – 42; #1 – 6 all, 8 – 34 even.