1. 2.4 – Linear Inequalities in One Variable
An inequality is a statement that contains one of the symbols: < , >, ≤ or ≥.
Equations
Inequalities
x = 3
x > 3
12 = 7 – 3y
12 ≤ 7 – 3y
A solution of an inequality is a value of the variable that makes the inequality a true statement.
The solution set of an inequality is the set of all solutions.

2. 2.4 – Linear Inequalities in One Variable

3. 2.4 – Linear Inequalities in One Variable

4. 2.4 – Linear Inequalities in One Variable
Example
Graph each set on a number line and then write it in interval notation. a. b. c.

5. 2.4 – Linear Inequalities in One Variable
Addition Property of Inequality
If a, b, and c are real numbers, then
a < b and a + c < b + c
a > b and a + c > b + c
are equivalent inequalities.
Also,
If a, b, and c are real numbers, then
a < b and a - c < b - c
a > b and a - c > b - c
are equivalent inequalities.

6. 2.4 – Linear Inequalities in One Variable
Example
Solve: Graph the solution set. [

7. 2.4 – Linear Inequalities in One Variable
Multiplication Property of Inequality
If a, b, and c are real numbers, and c is positive, then
a < b and ac < bc
are equivalent inequalities.
If a, b, and c are real numbers, and c is negative, then
a < b and ac > bc
are equivalent inequalities.
The direction of the inequality sign must change when multiplying or dividing by a negative value.

8. 2.4 – Linear Inequalities in One Variable
Example
Solve: Graph the solution set.
The inequality symbol is reversed since we divided by a negative number. (

9. 2.4 – Linear Inequalities in One Variable
Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5(x – 1)
3x + 9 ≥ 5x – 5
3x – 3x + 9 ≥ 5x – 3x – 5
9 ≥ 2x – 5
9 + 5 ≥ 2x – 5 + 5
14 ≥ 2x
7 ≥ x
x ≤ 7 [

10. 2.4 – Linear Inequalities in One Variable
Example
Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set.
7(x – 2) + x > –4(5 – x) – 12
7x – 14 + x > –20 + 4x – 12
8x – 14 > 4x – 32
8x – 4x – 14 > 4x – 4x – 32
4x – 14 > –32
4x – > –
4x > –18
x > –4.5 (

11. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Intersection of Sets
The solution set of a compound inequality formed with and is the intersection of the individual solution sets.

12. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Find the intersection of:
The numbers 4 and 6 are in both sets.
The intersection is {4, 6}.

13. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Solve and graph the solution for x + 4 > 0 and 4x > 0.
First, solve each inequality separately.
x + 4 > 0
4x > 0
and
x > – 4
x > 0 (
(0, )

14. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
0  4(5 – x) < 8
0  20 – 4x < 8
0 – 20  20 – 20 – 4x < 8 – 20
– 20  – 4x < – 12
Remember that the sign direction changes when you divide by a number < 0! ( 3 4 5 [
5  x > 3
(3,5]

15. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example – Alternate Method
0  4(5 – x) < 8
0  4(5 – x)
4(5 – x) < 8
0  20 – 4x
20 – 4x < 8
0 – 20  20 – 20 – 4x
20 – 20 – 4x < 8 – 20
– 20  – 4x
– 4x < – 12
Dividing by negative:
change sign
Dividing by negative:
change sign
x > 3
5  x ( 3 4 5 [
(3,5]

16. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Union of Sets
The solution set of a compound inequality formed with or is the union of the individual solution sets.

17. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Find the union of:
The numbers that are in either set are
{2, 3, 4, 5, 6, 8}.
This set is the union.

18. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example: Solve and graph the solution for
5(x – 1)  –5 or 5 – x < 11
5(x – 1)  –5 or
5 – x < 11
5x – 5  –5
–x < 6
5x  0
x > – 6
x  0
(–6, )

19. Compound Inequalities
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example: or

20. 2.4 – Linear Inequalities in One Variable