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5.6 Linear inequalities in one variable

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter, the emphasis is on the mechanics of equation solving, which clearly explains how to isolate a variable. The goal is to help the student feel more comfortable with solving applied problems. Ample opportunity is provided for the student to practice translating words to symbols, which is an important part of the "Five-Step Method" of solving applied problems (discussed in modules (<link document="m21980"/>) and (<link document="m21979"/>)). Objectives of this module: understand the meaning of inequalities, be able to recognize linear inequalities, know, and be able to work with, the algebra of linear inequalities and with compound inequalities.

Overview

  • Inequalities
  • Linear Inequalities
  • The Algebra of Linear Inequalities
  • Compound Inequalities

Inequalities

Relationships of inequality

We have discovered that an equation is a mathematical way of expressing the relationship of equality between quantities. Not all relationships need be relationships of equality, however. Certainly the number of human beings on earth is greater than 20. Also, the average American consumes less than 10 grams of vitamin C every day. These types of relationships are not relationships of equality, but rather, relationships of inequality .

Linear inequalities

Linear inequality

A linear inequality is a mathematical statement that one linear expression is greater than or less than another linear expression.

Inequality notation

The following notation is used to express relationships of inequality:
> Strictly   greater   than < Strictly   less   than ≥ Greater   than   or   equal   to ≤ Less   than   or   equal   to

Note that the expression x > 12 has infinitely many solutions. Any number strictly greater than 12 will satisfy the statement. Some solutions are 13, 15, 90, 12.1 ,   16.3   and   102.51 .

Sample set a

The following are linear inequalities in one variable.

  1. x ≤ 12
  2. x + 7 > 4
  3. y + 3 ≥ 2 y − 7
  4. P + 26 < 10 ( 4 P − 6 )
  5. 2 r − 9 5 > 15
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The following are not linear inequalities in one variable.

  1. x 2 < 4 .
    The term x 2 is quadratic, not linear.
  2. x ≤ 5 y + 3 .
    There are two variables. This is a linear inequality in two variables.
  3. y + 1 ≠ 5 .
    Although the symbol ≠ certainly expresses an inequality, it is customary to use only the symbols < ,   > ,   ≤ ,   ≥ .
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Practice set a

A linear equation, we know, may have exactly one solution, infinitely many solutions, or no solution. Speculate on the number of solutions of a linear inequality. ( Hint: Consider the inequalities x < x − 6 and x ≥ 9 .)

A linear inequality may have infinitely many solutions, or no solutions.

The algebra of linear inequalities

Inequalities can be solved by basically the same methods as linear equations. There is one important exception that we will discuss in item 3 of the algebra of linear inequalities.

The algebra of linear inequalities

Let a ,   b ,   and   c represent real numbers and assume that
a < b ( or   a > b )
Then,   if   a < b ,

  1. a + c < b + c and a − c < b − c .
    If any real number is added to or subtracted from both sides of an inequality, the sense of the inequality remains unchanged.
  2. If c is a positive real number, then if a < b ,
    a c < b c and a c < b c .
    If both sides of an inequality are multiplied or divided by the same positive number the sense of the inequality remains unchanged.
  3. If c is a negative real number, then if a < b ,
    a c > b c and a c > b c .
    If both sides of an inequality are multiplied or divided by the same negative number, the inequality sign must be reversed (change direction) in order for the resulting inequality to be equivalent to the original inequality. (See problem 4 in the next set of examples.)

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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