College Algebra Lecture Notes Section 1.2Page 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:
  • Set notation
  • Interval notation (recall that brackets [] mean to include the endpoints and parentheses mean ( ) mean that the endpoints are not included
  • Number line graph

Comparison of the Representations of an Inequality
Inequality / Set Notation / Interval Notation / Number Line
x is greater than 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 /
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

AB, then

A + CB + 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
AB, then
ACBC,
if C is a positive quantity (i.e., C > 0)
Summary: Like positive quantities can multiply both sides of an inequality without changing the inequality. / If A, B, and C represent algebraic expressions, and
AB, then
ACBC,
if C is a negative quantity (i.e., C < 0)
Summary: If like negative quantities multiply both sides of an inequality, then the inequality reverses.

Practice:

  1. Solve the inequality and state the solution using all three representations.

  1. Solve the inequality and state the solution using all three representations.

C. SOLVING COMPOUND INEQUALITIES

  • Compound Inequality:when the range of desired solutions must obey more than one inequality at once.
  • Example: If you want to be in the second tax bracket, then you must make $8,350 or more butless than $33,950
     income $8,350 and income $33,950
    $8,350  income $33,950
  • 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.”

Practice:

  1. Solve the compound inequality and state the solution using all three representations.
  1. Solve the compound inequality and state the solution using all three representations. Also write the solution using compound inequality notation.

D. APPLICATIONS OF INEQUALITIES

  • Writing the domain of functions

Practice:

  1. State the domain of the expression , , and .
  • Any real-world problem where a range of values is acceptable for an answer.

Practice:

  1. 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?