2.4 – Linear Inequalities in One Variable
An inequality is a statement that contains one of the
symbols: < , >, ≤ or ≥.
Equations Inequalities
x = 3
12 = 7 – 3y
x > 3
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.
2.4 – Linear Inequalities in One Variable
2.4 – Linear Inequalities in One Variable
Example
Graph each set on a number line and then write it in
interval notation. a. { | 2}
b. { | 1}
c. { | 0.5 3}
x x
x x
x x
a. [2, )
b.
c. (0.5,3]
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.
2.4 – Linear Inequalities in One Variable
Example
Solve: Graph the solution set. 3 4 2 6x x
{ | 10} or 10,x x
[
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.
2.4 – Linear Inequalities in One Variable
The direction of the inequality sign must change
when multiplying or dividing by a negative value.
Solve: Graph the solution set.
{ | 3} or 3,x x
2.3 6.9x
The inequality symbol is reversed since we divided by a
negative number.
(
Example
2.4 – Linear Inequalities in One Variable
Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set.
3x + 9 ≥ 5x – 5
3x – 3x + 9 ≥ 5x – 3x – 5
9 ≥ 2x – 5
14 ≥ 2x
7 ≥ x
9 + 5 ≥ 2x – 5 + 5
3x + 9 ≥ 5(x – 1)
x ≤ 7
[
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 – 14 + 14 > –32 + 14
4x > –18
x > –4.5
(
2.4 – Linear Inequalities in One Variable
Intersection of Sets
The solution set of a compound inequality formed with
and is the intersection of the individual solution sets.
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example
Find the intersection of: {2,4,6,8} {3,4,5,6}
The numbers 4 and 6 are in both sets.
The intersection is {4, 6}.
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Solve and graph the solution for x + 4 > 0 and 4x > 0.
Example
First, solve each inequality separately.
x + 4 > 0
x > – 4
4x > 0
x > 0 and
-4
(0
(
( (0, )
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
5 x > 3
Remember that the sign direction
changes when you divide by a
number < 0!
(
[
(3,5]
2.4 – Linear Inequalities in One Variable
3 4 5
Compound Inequalities
Example – Alternate Method
0 4(5 – x)
0 20 – 4x
0 – 20 20 – 20 – 4x
– 20 – 4x
5 x
(
[
(3,5]
2.4 – Linear Inequalities in One Variable
3 4 5
4(5 – x) < 8
20 – 4x < 8
20 – 20 – 4x < 8 – 20
– 4x < – 12
x > 3
0 4(5 – x) < 8
Dividing by negative:
change sign
Dividing by negative:
change sign
Compound Inequalities
The solution set of a compound inequality formed with or is
the union of the individual solution sets.
Union of Sets
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Find the union of:
Example
{2,4,6,8} {3,4,5,6}
The numbers that are in either set are
{2, 3, 4, 5, 6, 8}.
This set is the union.
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
5x – 5 –5
5x 0
x 0
5 – x < 11
–x < 6
x > – 6
or
0
[-6
(
(–6, ) -6
(
2.4 – Linear Inequalities in One Variable
Compound Inequalities
Example:
or
,
2.4 – Linear Inequalities in One Variable
Compound Inequalities
2.4 – Linear Inequalities in One Variable
Top Related