10.2: Solving Inequalities
I can transform inequalities in order to solve them.
Important Reminders about Inequalities
1. a < 2 reads “a is less than 2.”
Open circle at 2, shaded to the left.
2. a ≤ 2 reads “a is less than or equal to 2.”
Closed circle at 2, shaded to the left.
3. a > 2 reads “a is greater than 2.”
Open circle at 2, shaded to the right.
4. a ≥2 reads “a is greater than or equal to2.”
Closed circle at 2, shaded to the right.
Property of Comparison: for all real numbers a and b, one and only one of the following statements is true:
a < b,
a = b,
or
a>b
Suppose you know two facts about the graphs of three numbers a, b, and c:
1. The graph of a is to the left of the graph of b.
2. The graph of b is to the left of the graph of c.
a
b
c
From the graphs above, you can see that the graph of a is to the left of the graph of c; therefore,
a < c, which illustrates the Transitive Property of Order.
Transitive Property of Order: For all real numbers a, b, and c:
1. If a < b and b < c, then a < c.
2. If c > b and b > a, then c > a.
The following properties guarantee that the transformations of a given inequality always produce equivalent
inequalities, that will have the same solution set as the original inequality:
Addition Property of Order (Addition Property of Inequality): If you add the same number to each side
of an inequality, the inequality remains true.
Ex. #1 (p.466)
y–2 ≥ 7
+2
+2
y ≥ 9
Graph D (Closed circle at 9, shaded to the right)
10.2: Solving Inequalities
I can transform inequalities in order to solve them.
Subtraction Property of Inequality: If you subtract the same number from each side of an inequality, the
inequality remains true.
Ex. #9 (p.466)
b–1 < b–2
+1
+1
b < b -1
-b
-b I
0 < -1
False, as 0 is not less than -1.
Graph F (No Solution, so there is no graph)
Multiplication Property of Order (Multiplication Property of Inequality) – Case 1: If you multiply
each side of an inequality by the same positive number, the inequality remains true.
𝒅
Ex. #6 (p.466)
𝟐
2 X
𝑑
2
> −𝟏𝟎
> -10
X 2
d > -20
Graph G (Open Circle at -20, shaded to the right)
Division Property of Inequality – Case 1: If you divide each side of an inequality by the same positive
number, the inequality remains true.
Ex. #3 (p.466)
6p < 24
6
6
p < 4
Graph J (Open Circle at 4, shaded to the left)
Multiplication Property of Order (Multiplication Property of Inequality) – Case 2: If you multiply
each side of an inequality by the same negative number, the direction of inequality symbol must be
reversed for the inequality to remain true.
Ex. #21 (p.466)
𝒅
≥ 𝟎
𝑑
≥ 0 ∗ (-3)
− 𝟑
(−3) ∗ − 3
𝑑
≤
0
Since we are MULTIPLYING both sides by (-3),
we must REVERSE the direction of the inequality.
Division Property of Inequality – Case 2: If you divide each side of an inequality by the same negative
number, the direction of inequality symbol must be reversed for the inequality to remain true.
Ex. #18 (p.466)
-5w ≤ 15
-5
-5
w ≥ -3
Since we are DIVIDING both sides by (-5), we must
REVERSE the direction of the inequality.
10.2: Solving Inequalities
I can transform inequalities in order to solve them.
To Solve Multi-Step Inequalities:
1. Simplify each side of the inequality, as needed (grouping symbols).
2. Use the inverse operation to undo any addition or subtraction.
3. Use the inverse operation to undo any multiplication or division.
Ex. #36 (p.467)
7a
7a
7a
- 7a
0
< 3 + 7 (a – 1)
< 3 + 7a – 7
< 7a – 4
- 7a
d
< -4
False, as 0 is not less than – 4; therefore
No Solution, so there is no graph.