Section 2.7 Solving Linear Inequalities
Linear Inequality in One Variable:
A linear inequality in one variable is an inequality that can be written in the form ax + b < c where a, b, and c
are real numbers and a ≠ 0. This also holds true for the inequality symbols >, ≤, and ≥.
Words
Less Than
Inequality Sign
Number Line Dot
<
Open dot
○
Parentheses
()
Interval Notation
Less Than or
Equal To
≤
Closed dot
●
Brackets
[]
Greater Than
Greater Than or
Equal To
≥
Closed dot
●
Brackets
[]
>
Open dot
○
Parentheses
()
Solutions to Inequalities in inequality form:
Review:
< - means “less than”
≤ - means “greater than”
> - means “less than or equal to”
≥ - means “greater than or equal to”
To find the solution of an inequality, we solve for x. This means we will have to get x by itself.
Examples: x > 3
x≤5
means the value of x is anything greater than 3.
means the value of x is anything less than or equal to 5.
Graphing solutions of inequalities:
To graph the solutions of inequalities, we will use a number line. For solutions where x is < or >, we will use an
open dot ○ on the number. For solutions where x is ≤ or ≥, we will use a closed dot ● on the number. From
there we draw an arrow from the dot to represent all values of x.
x≤5
Examples: x > 3
-3 -2
-1 0
1
2
3
4 5
-3
x≥4
-2 -1 0
1
2
3
4
5
6
1
2
3
4
5
6
x < -1
-3 -2 -1 0
1
2
3
4
5
6
-3
-2 -1 0
Writing Solutions in Interval Notation:
To write a solution for x in interval notation, we use parentheses and/or brackets. If x is greater than or greater
than or equal to a number, it goes towards infinity ∞. If x is less than or less than or equal to a number, it goes
towards negative infinity -∞. We will always use parentheses on infinity and negative infinity.
Examples:
x>3
(3, ∞)
x≤5
(-∞, 5]
x≥4
[4, ∞)
x < -1
(-∞, -1)
Addition Property of Inequality:
If a, b, and c are real numbers, then a < b and a + c < b + c are equivalent inequalities.
Example 1: Solve x  4  6 .
x  4  6
x  4  4  6  4
x  10
(, 10]
Multiplication Property of Inequality:
1. If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities.
2. If a, b, and c are real numbers, and c is negative, then a < b and ac > bc are equivalent inequalities.
What this means: When you multiply or divide by a negative while solving an inequality, you have to
flip/change/reverse the inequality sign.
Example 2: Solve 2x  4 .
Example 3: Solve 2x  4 .
2 x  4
2 x 4

2 2
x2
2 x  4
2 x 4

2
2
x  2
Example 4: Solve, graph, and write in interval notation: 4x  7  9 .
4 x  7  9
4 x  7  7  9  7
4 x  16
-3 -2
-1 0
1
2
3
4 5
(-∞, 4]
4 x 16

4
4
x4
Example 5: Solve, graph, and write in interval notation: 5 x  7  2( x  3)
5 x  7  2( x  3)
-3 -2
-1 0
1
2
3
4 5
5 x  7  2 x  6
5 x  7  2 x  2 x  6  2 x
7 x  7   6
7 x  7  7   6  7
7 x  13
7 x 13

7
7
13
x
7
 13 
 ,
7

Compound Inequalities:
Inequalities containing one inequality symbol are called simple inequalities, while inequalities containing two
inequality symbols are called compound inequalities. A compound inequality is two simple inequalities
combined:
3<x<5
means 3 < x and x < 5
Read: “x is greater than three and less than five”.
Example 6: Solve, graph, and write in interval notation: 1  2x  3  5 .
1  2 x  3  5
1  3  2 x  3  3  5  3
-3
-2 -1 0
1
2
3
4 5
2  2x  8
1, 4)
2 2x 8


2 2 2
1 x  4
Using inequalities with word problems:
Example 7: Mary has $350 to spend on clothes (exclude taxes). She plans on buying three pairs of jeans
priced at $50 each and then several t-shirts priced at $15 dollars each. How many t-shirts can Mary buy along
with the three pairs of jeans and stay within her budget?
x = the number of t-shirts
3(50)  15 x  350
Solution: x ≤13.33, which means Mary can buy 13 t-shirts and three pairs of jeans for less than $350.
Example 8: Danny works at a local car dealership. He earns $2,000 a month plus 5% commission of the cars
he sells. What is the value of the cars Danny must sell in a year to earn at least $65,000?
x = the value of the cars
12(2, 000)  0.05 x  65, 000
Solution: x ≥ $820,000, which means Danny must sell $820,000 worth of cars during a year to earn at least
$65,000 a year.