Glossary for Lesson 8: Inequalities and Absolute Value

Number Line

The real number line is a horizontal line used to represent the real numbers.  Every point on the line represents a different real number, with values increasing from left to right.  The arrows on either end indicate that the line extends forever in either direction, which allows it to include all values from -\inf to \inf

Inequality

Whereas an equation relates two quantities that are of equal value, an inequality is a way to mathematically relate two quantities that do not have the same value.  An inequality may use one of the following symbols:

  • a\neq b means that a is not equal to b .
  • a>b means that a is greater than b .
  • a<b means that a is less than b .
  • a\ge b means that a is greater than or equal to b .
  • a\le b means that a is less than or equal to b . The set of all numbers that satisfy an inequality (i.e. can be plugged in for the variables to make the inequality true) is called the solution set of that inequality.  Solution sets are often represented by graphs on number lines.
    To write \neq with MathQuill, type \neq and then a space.  To write \le, type \le and then a space, and to write \ge , type \ge and then a space.

Chained Inequality

If we know that a<b and that b<c , then we can use a chained inequality or chained notation to combine these two inequalities into one statement: a<b<c

Interval

An interval is a set of real numbers that includes all real numbers between a lower bound and an upper bound.  If an interval represents a set of values that a variable, let’s say x for now, is allowed to equal, then it might be expressed with inequality notation: a<x<b , a\le x<b , a<x\le b , or a\le x\le b Intervals can also be expressed using what is called interval notation.  For the intervals written above, we could instead use interval notation to write (a,b) , [a,b) , (a,b] , or [a,b] .  Interval notation thus uses square or round brackets surround the lower and upper bound of the interval, separated by a comma.  A round bracket or parenthesis indicates that the bound or endpoint next to it is not included in the interval, while a square bracket means that that number is included.
Let’s take a look at an example.  If we have -11\le x<4 , we could also write that x lies in the interval [-11,4) .  Here, x can equal -11 , since that has a square bracket next to it, but it can’t equal 4 , since that is by a round parenthesis.

Union

The union of two sets is the set of all values that belong to either of those two sets, and we denote the union of two sets A and B by writing A\union B .  So if we have A={-4,-2,3,5,6} and B={-3,0,3,4,5}, then A\union B={-4,-3,-2,0,3,5,6}.  In this course, we will most often use unions to join together intervals that make up solution sets.

Compound Inequality

A compound inequality connects two or more inequalities by using the words “and” or “or”.  In this course, we’ll limit ourselves to compound inequalities with just two inequalities.

  • If the word “and” is used, then a value must satisfy both of the inequalities in order to be in the solution set.  Example: If we start out with x-1>3 and x+2\le9 , then we can simply to have x>4 and x\le7 , so the solution is 4<x\le 7 , written as a chained inequality.  In interval notation, this is (4,7] .  Some values that would satisfy this compound inequality, then, are 5.6 , 8 , and 7 .  Some values that would not satisfy it are -1 , 4 , and 100 .
  • If the word “or” is used, then a value may satisfy either or both of the inequalities in order to be in the solution set.  Example: Let’s say we begin with 2x<12 or x-5>6 .  We could also write this as 2x<12 \or x-5>6 , where \or, typed as \or in MathQuill, just means “or”.  We can then simplify to have x<6 \or x>11.  In interval notation, we would write this as (-\inf ,6)\union (11,\inf ).  Some values that would satisfy this compound inequality, then, are -1796.45 , 0 , and 83 .  Some values that would not satisfy it are 6 , 8\frac {2}{3} , and 10 .

Absolute Value |a|

The absolute value |a| is the magnitude of a, that is, the number with any negative sign removed. For example, |3|=3 and |-3|=3. Note that any calculation within the absolute value is done first and then the sign ignored eg |5-8| = |-3| = 3.