Chapter 3, “Inequalities” from Beginning and Intermediate Algebra by Tyler Wallace is available under a Creative Commons Attribution 3.0 Unported license. © 2010.

3.1Inequalities - Solve and Graph InequalitiesObjective: Solve, graph, and give interval notation for the solution tolinear inequalities.When we have an equation such asx= 4we have a specific value for our variable.With inequalities we will give a range of values for our variable. To do this wewill not use equals, but one of the following symbols:>Greater thangreaterorequalslantGreater than or equal to<Less thanlessorequalslantLess than or equal toWorld View Note:English mathematician Thomas Harriot first used the abovesymbols in 1631. However, they were not immediately accepted as symbols suchas⊏and⊐were already coined by another English mathematician, WilliamOughtred.If we have an expression such asx <4, this means our variable can be any numbersmaller than 4 such as−2,0,3,3.9 or even 3.999999999 as long as it is smaller118

Example 151.Graph the inequality and give the interval notationx <2Start at2and shade belowUse)for less thanOur Graph(− ∞,2)Interval NotationExample 152.Graph the inequality and give the interval notationygreaterorequalslant−1Start at−1and shade aboveUse[for greater than or equalOur Graph[−1,∞)Interval NotationWe can also take a graph and find the inequality for it.119

Example 153.Give the inequality for the graph:Graph starts at 3 and goes up or greater. Curved bracket means just greater thanx >3Our Solution