Inequalities: Solve Using Addition and Subtraction
Tool Box:
Summary:
Equivalent Inequalities
Properties of Inequalities
Closed Circle
Open Circle
Question:
Solving an Inequality: means finding values 𝑠 − 7 < 5, s = 14
for the variable that make the inequality true 14 − 7 < 5
7
<5
7 ≰5
𝑥
Graph of a Linear Inequality in One
12 ≥ 𝑎 + 2, a = 10
12 ≥ 10 + 2
12 ≥ 12
>
5
Variable: The set of points on a number line that
represent “all” solutions of the inequality
Equivalent Inequalities: Inequalities that have
the same solution(s)
INEQUALITY SYMBOLS
<
>
Less than
Fewer than
≤
Greater than
More than
Exceeds
≥
At most
No more than
Less than or
or equal to
At least
No less than
Greater than
or equal to
Solve an Inequality by Adding
EXAMPLE:
𝒕 −
𝒕 −
𝟒𝟓 ≤
𝟒𝟓 ≤
𝟏𝟑
𝟏𝟑
𝒕 − 𝟒𝟓 ≤
𝟏𝟑
𝒕 − 𝟒𝟓 ≤
𝟏𝟑
+ 𝟒𝟓
+ 𝟒𝟓
𝑡
≤
58
This means all numbers less than or equal to 58
are solutions to the inequality
47
49
51
53
The heavy red arrow pointing to the left
shows that the inequality includes “all”
real numbers less than 58”
55
57
1.Write the inequality
2. Draw railroad tracks
3. Isolate the variable “t”
4. Addition Property of Inequality
(Add 45 to both sides of the inequality)
5. Simplify/Combine Like Terms
The solution can also be represented on
a number line.
58 59
61
63
65
The “closed” red circle means that 58 is
included as part of the solution to the
inequality
Another way to write a solution set
Read t “such that” t is less than or
equal to 58
SET-BUILDER NOTATION:
{𝑡 | 𝑡 ≤ 58}
Solve and Graph an Inequality
EXAMPLE
𝟔 < 𝒙 − 𝟑. 𝟑
𝟔
Solve for “x”
> 𝒙 − 𝟑. 𝟑
1. Write the inequality
2. Draw railroad tracks
3. Isolate the variable “x”
4. Addition Property of Inequality
(add “3.3” to both sides of the
inequality
5. Combine Like Terms/Simplify
𝟔 > 𝒙 − 𝟑. 𝟑
+ 𝟑. 𝟑
+ 𝟑. 𝟑
9.3
9.3
8
< 𝑥
The solution set is {𝑥| 𝑥 > 9.3 }
< 𝑥 is the same as 𝑥 > 9.3
8.1 8.2 8.3 8.4 8.5
8.7
8.9
The “open” red circle at 9.3 indicates
that 9.3 is not included in the solution
of the inequality
9
9.1 9.2 9.3
Solve for “r”
𝟏𝟗 + 𝒓 ≥ 𝟏𝟔
EXAMPLE:
𝟏𝟗 + 𝒓 ≥ 𝟏𝟔
1. Write the inequality
2. Draw railroad tracks
2. Isolate the variable “r”
3. Subtraction Property of Inequality
(Subtract 19 from both sides of the
inequality)
4. Combine Like Terms/Simplify
Set Builder Notation
𝟏𝟗 + 𝒓 ≥ 𝟏𝟔
𝟏𝟗 + 𝒓 ≥ 𝟏𝟔
−𝟏𝟗
− 𝟏𝟗
𝑟 ≥ −3
{𝒓 | 𝒓 ≥ − 𝟑}
-7
9.7
The heavy red arrow pointing to the right shows
that the inequality includes all numbers greater
than 9.3
Solve an Inequality by Subtracting
-9
9.5
-5
-3
-1
0
1
Solve an Inequality with Variables
on Both Sides
3
5
7
Solve for “p”
EXAMPLE:
5𝑝 + 7 >
6𝑝
1 Write the equation
2 Draw railroad tracks
9
5𝑝 + 7 >
5𝑝 + 7 >
−𝟓𝒑
6𝑝
3. Isolate the variable. Since 6p is on
the right side and the constant 7 is on
the left, move the 5p to the right side of
the equation to combine like terms
4. Subtraction Property of Inequality
(subtract 5p from both sides of the
inequality)
5. Combine Like Terms/Simplify
Set Builder Notation
6𝑝
− 𝟓𝒑
7 ˃
p
{𝑝 | 𝑝 < 7}
Write an Inequality to Solve a
Problem
Olympics: Yulia Raskina scored a total of 39,548 points in four events of rhythmic
gymnastics. Yulia Barsukova scored 9,883 in rope competition, 9,900 points in the hoop
competition, and 9,916 in the ball competition. How many points did Barsukova need to
score in the ribbon competition to surpass Raskina and win the GOLD Medal?
Words: Barsukova total points must be greater than (>) Raskina’s total points
Variable: Let r = Barsukova’s score in the ribbon competition
Inequality: 9,883 + 9,900 + 9,916 + r
{
}
𝐵𝑎𝑟𝑠𝑢𝑘𝑜𝑣𝑎′ 𝑠 𝑇𝑜𝑡𝑎𝑙
>
Greater than
39,548
{𝑅𝑎𝑠𝑘𝑖𝑛𝑎′ 𝑠 𝑇𝑜𝑡𝑎𝑙}
Solve the Inequality
9,883 + 9,90 + 9,916 + r >
29,669
29,669
29,669
- 29, 669
+𝑟 >
+ 𝑟 >
+ 𝑟 >
-
39,548
39,548
39,548
39,548
29,669
𝑟 > 9,849
Barsukova needed to score more than 9,849 points
to win the gold medal
1. Write the inequality
2. Train Tracks
3. Combine Like Terms/Simplify
4. Isolate the variable “r”
5. Subtraction Property of Inequality
(subtract 29,669 from each side of the
inequality)
6. Combine Like Terms/Simplify