Section 1.2 - Linear Inequalities

Section Objectives

  1. Write inequalities corresponding to problem situations.
  2. Write an interval using inequalities and graph it.
  3. Solve linear inequalities.
  4. Solve application problems involving linear inequalities.



A inequality is an algebraic expression involving less than (), less than or equal to (), greater than (), or greater than or equal to ().



Using Inequalities to Describe Intervals

Inequalities are often used to describe intervals. Here are some examples.

ConditionsSet NotationInterval Notation
is greater than
is less than or equal to
is less than and greater than
is less than and greater than or equal to
is less than or is greater than

 

The last three examples are called compound inequalities. Compound inequalities are associated with the words "and" or "or". "And" inequalities describe intersections of sets, whereas "or" inequalities describe unions. We use the symbol to denote a union, and the symbol to denote an intersection.


To graph an inequality, we shade the interval(s) along a number line. Use parentheses ( ) and open dots to exclude endpoints. Use brackets [ ] and closed (filled-in) dots to include endpoints.



Examples













Number line shading numbers less than or equal to -4










Number line shading numbers less than 2, as well as numbers greater than or equal to 4




Solving inequalities

To solve an inequality means to find ALL replacements for the variable that make the inequality true. Your answers will typically (but not always) be intervals.


To solve a linear inequality, we apply the same process as we used for solving linear equations, with one exception: When solving a linear inequality, whenever you multiply (or divide) across by a negative number, you must reverse the inequality symbol(s).



Examples

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For further help...

  1. Khan Academy has excellent videos on solving linear inequalities. Start at this page.
  2. Do a Google search for "solving one-variable linear inequalities".