2.7 Linear inequalities and absolute value inequalities

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In this section you will:
• Use interval notation.
• Use properties of inequalities.
• Solve inequalities in one variable algebraically.
• Solve absolute value inequalities.

It is not easy to make the honor role at most top universities. Suppose students were required to carry a course load of at least 12 credit hours and maintain a grade point average of 3.5 or above. How could these honor roll requirements be expressed mathematically? In this section, we will explore various ways to express different sets of numbers, inequalities, and absolute value inequalities.

Using interval notation

Indicating the solution to an inequality such as $\text{\hspace{0.17em}}x\ge 4\text{\hspace{0.17em}}$ can be achieved in several ways.

We can use a number line as shown in [link] . The blue ray begins at $\text{\hspace{0.17em}}x=4\text{\hspace{0.17em}}$ and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.

We can use set-builder notation : $\text{\hspace{0.17em}}\left\{x|x\ge 4\right\},$ which translates to “all real numbers x such that x is greater than or equal to 4.” Notice that braces are used to indicate a set.

The third method is interval notation    , in which solution sets are indicated with parentheses or brackets. The solutions to $\text{\hspace{0.17em}}x\ge 4\text{\hspace{0.17em}}$ are represented as $\text{\hspace{0.17em}}\left[4,\infty \right).\text{\hspace{0.17em}}$ This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.

The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be “equaled.” A few examples of an interval    , or a set of numbers in which a solution falls, are $\text{\hspace{0.17em}}\left[-2,6\right),$ or all numbers between $\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}6,$ including $\text{\hspace{0.17em}}-2,$ but not including $\text{\hspace{0.17em}}6;$ $\left(-1,0\right),$ all real numbers between, but not including $\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0;$ and $\text{\hspace{0.17em}}\left(-\infty ,1\right],$ all real numbers less than and including $\text{\hspace{0.17em}}1.\text{\hspace{0.17em}}$ [link] outlines the possibilities.

Set Indicated Set-Builder Notation Interval Notation
All real numbers between a and b , but not including a or b $\left\{x|a $\left(a,b\right)$
All real numbers greater than a , but not including a $\left\{x|x>a\right\}$ $\left(a,\infty \right)$
All real numbers less than b , but not including b $\left\{x|x $\left(-\infty ,b\right)$
All real numbers greater than a , including a $\left\{x|x\ge a\right\}$ $\left[a,\infty \right)$
All real numbers less than b , including b $\left\{x|x\le b\right\}$ $\left(-\infty ,b\right]$
All real numbers between a and b , including a $\left\{x|a\le x $\left[a,b\right)$
All real numbers between a and b , including b $\left\{x|a $\left(a,b\right]$
All real numbers between a and b , including a and b $\left\{x|a\le x\le b\right\}$ $\left[a,b\right]$
All real numbers less than a or greater than b $\left\{x|xb\right\}$ $\left(-\infty ,a\right)\cup \left(b,\infty \right)$
All real numbers $\left(-\infty ,\infty \right)$

Using interval notation to express all real numbers greater than or equal to a

Use interval notation to indicate all real numbers greater than or equal to $\text{\hspace{0.17em}}-2.$

Use a bracket on the left of $\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ and parentheses after infinity: $\text{\hspace{0.17em}}\left[-2,\infty \right).$ The bracket indicates that $\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ is included in the set with all real numbers greater than $\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ to infinity.

Use interval notation to indicate all real numbers between and including $\text{\hspace{0.17em}}-3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}5.$

$\left[-3,5\right]$

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