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College Algebra Lecture Notes
Section 1.2
Page 1 of 5
Section 1.2: Linear Inequalities in One Variable
Big Idea: An inequality is used when a range of numbers are desired as the result of a
calculation.
Big Skill: You should be able to solve linear inequalities using the Properties of Inequalities.
Examples of when you might want to use an inequality:
 To get an A in this class, you must get between 92% and 100%  0.92  score  1.00
 You can only afford a car that costs less than $4,000  price < $4,000
 To escape the earth’s gravity, a rocket’s launch speed must be more than 25,000 mph 
speed > 25,000 mph
A. INEQUALITIES AND SOLUTION SETS
 While conditional linear equations in one variable have one solution, linear inequalities
usually have an infinite number of solutions (especially when the answers can be real
numbers)
 Solution set: the set of all numbers that satisfy an inequality
 Three ways to represent a solution set:
o Set notation
o Interval notation (recall that brackets [] mean to include the endpoints and
parentheses mean ( ) mean that the endpoints are not included
o Number line graph
Inequality
x is greater than k
Comparison of the Representations of an Inequality
Set Notation Interval Notation
Number Line
x   k, 
 x | x  k
x is greater than or
equal to k
x is less than k
x is less than or equal
to k
x is greater than a and
less than b
x is greater than or
equal to a and less
than b
x is greater than or
equal to a and less
than or equal to b
x is greater than a and
less than or equal to b
College Algebra Lecture Notes
Section 1.2
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x is less than a or
greater than b
x is less than or equal
to a or greater than b
x is less than or equal
to a or greater than or
equal to b
x is less than a or
greater than or equal
to b
B. SOLVING LINEAR INEQUALITIES
 The usual way to solve a linear inequality is to keep re-writing the inequality as a
sequence of equivalent inequality, each one simpler than the one before.
 Inequalities can be shown to be equivalent by using the Properties of Inequality:
The Additive Property of Inequality
If A, B, and C represent algebraic expressions, and
A < B, then
A + C < B + C.
Summary: Like quantities can be can be added (or subtracted) to both sides of an inequality
without changing the inequality.
The Multiplicative Property of Equality
If A, B, and C represent algebraic expressions,
and
A < B, then
AC < BC,
if C is a positive quantity (i.e., C > 0)
If A, B, and C represent algebraic expressions,
and
A < B, then
AC > BC,
if C is a negative quantity (i.e., C < 0)
Summary: Like positive quantities can
multiply both sides of an inequality without
changing the inequality.
Summary: If like negative quantities multiply
both sides of an inequality, then the inequality
reverses.
Practice:
1. Solve the inequality 8n  5  2n 12 and state the solution using all three
representations.
College Algebra Lecture Notes
2. Solve the inequality 
Section 1.2
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2
y
y   2 and state the solution using all three representations.
5
10
C. SOLVING COMPOUND INEQUALITIES
 Compound Inequality: when the range of desired solutions must obey more than one
inequality at once.
o Example: If you want to be in the second tax bracket, then you must make $8,350
or more but less than $33,950
 income  $8,350 and income < $33,950
 $8,350  income < $33,950
o Example: If you want to be avoid a parade route, you must be more than 5 blocks
east or west of block #22
 location > block #22 + 5 or location < block #22 - 5
 location > block #27 or location < block #17
 Since there is more than one inequality involved in a compound inequality, we will have
to figure out how to combine multiple inequalities. This can be done using the notions of
set unions and set intersection
 Intersection of two sets (i.e., A  B): A new set formed that includes only the elements
that are common to both sets. This is the first income tax example. Intersections always
got with the word “and.”
 Union of two sets (i.e., A  B): A new set formed that includes all elements that are in
either set. This is the second parade example. Unions always got with the word “or.”
College Algebra Lecture Notes
Section 1.2
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Practice:
3. Solve the compound inequality 3x 1  4 or 4x  3  6 and state the solution using
all three representations.
4. Solve the compound inequality 3x  5  13 and 3x  5  1 and state the solution using
all three representations. Also write the solution using compound inequality notation.
College Algebra Lecture Notes
Section 1.2
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D. APPLICATIONS OF INEQUALITIES
 Writing the domain of functions
Practice:
5. State the domain of the expression

24 x  2
,
, and 12  2x .
x x2
Any real-world problem where a range of values is acceptable for an answer.
Practice:
6. Say you’ve earned test scores of 78, 85, and 88 (out of 100), and you want your test score
average to be at least 86. What do you need to get on the fourth and final test to make
this happen?