Rational Inequalities
Rational Inequalities

Learning Outcomes

  • Represent inequalities using an inequality symbol
  • Represent inequalities on a number line

An inequality is a mathematical statement that compares two expressions using a phrase such as greater than or less than. Special symbols are used in these statements. In algebra, inequalities are used to describe sets of values, as opposed to single values, of a variable. Sometimes, several numbers will satisfy an inequality, but at other times infinitely many numbers may provide solutions. Rather than try to list a possibly infinitely large set of numbers, mathematicians have developed some efficient ways to describe such large lists.

Inequality Symbols

One way to represent such a list of numbers, an inequality, is by using an inequality symbol:

  • [latex]{x}\lt{9}[/latex] indicates the list of numbers that are less than [latex]9[/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[/latex].
  • [latex]-5\le{t}[/latex] indicates all the numbers that are greater than or equal to [latex]-5[/latex].

If you were to read the above statement from left to right, it would translate as [latex]-5[/latex] is less than or equal to t. The direction of the symbol is dependent upon the statement you wish to make. For example, the following statements are equivalent. Both represent the list of all numbers less than 9. Note how the open end of the inequality symbol faces the larger value while the smaller, pointy end points to the smaller of the values:

  • [latex]{x}\lt{9}[/latex]
  • [latex]{9}\gt{x}[/latex]

Here’s another way of looking at is:

  • [latex]x\lt5[/latex] means all the real numbers that are less than 5, whereas;
  • [latex]5\lt{x}[/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\gt{5}[/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.

The box below shows the symbol, meaning, and an example for each inequality sign, as they would be translated reading from left to right.

Symbol Words Example
[latex]\neq [/latex] not equal to [latex]{2}\neq{8}[/latex], 2 is not equal to 8.
[latex]\gt[/latex] greater than [latex]{5}\gt{1}[/latex], 5 is greater than 1
[latex]\lt[/latex] less than [latex]{2}\lt{11}[/latex], 2 is less than 11
[latex] \geq [/latex] greater than or equal to [latex]{4}\geq{ 4}[/latex], 4 is greater than or equal to 4
[latex]\leq [/latex] less than or equal to [latex]{7}\leq{9}[/latex], 7 is less than or equal to 9

The inequality [latex]x>y[/latex] can also be written as [latex]{y}<{x}[/latex]. The sides of any inequality can be switched as long as the inequality symbol between them is also reversed.

Graphing an Inequality

Another way to represent an inequality is by graphing it on a number line:

A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero.

Below are three examples of inequalities and their graphs. Graphs are often helpful for visualizing information.

[latex]x\leq -4[/latex]. This translates to all the real numbers on a number line that are less than or equal to [latex]4[/latex].

Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4.

[latex]{x}\geq{-3}[/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.

Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3.

Each of these graphs begins with a circle—either an open or closed (shaded) circle. This point is often called the end point of the solution. A closed, or shaded, circle is used to represent the inequalities greater than or equal to [latex] \displaystyle \left(\geq\right) [/latex] or less than or equal to [latex] \displaystyle \left(\leq\right) [/latex]. The end point is part of the solution. An open circle is used for greater than (>) or less than (<). The end point is not part of the solution. When the end point is not included in the solution, we often say we have strict inequality rather than inequality with equality.

The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \displaystyle x\geq -3[/latex] shown above, the end point is [latex]−3[/latex], represented with a closed circle since the inequality is greater than or equal to [latex]−3[/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]−3[/latex]. The arrow at the end indicates that the solutions continue infinitely.

Example

Graph the inequality [latex]x\ge 4[/latex]

We can use a number line as shown. Because the values for [latex]x[/latex] include [latex]4[/latex], we place a solid dot on the number line at [latex]4[/latex].

Then we draw a line that begins at [latex]x=4[/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to [latex]4[/latex].

A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.

This video shows an example of how to draw the graph of an inequality.

Example

Write an inequality describing all the real numbers on the number line that are strictly less than [latex]2[/latex]. Then draw the corresponding graph.

We need to start from the left and work right, so we start from negative infinity and end at [latex]2[/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[/latex].

Inequality: [latex]x\lt2[/latex]

To draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.

Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.

You are watching: College Algebra Corequisite. Info created by PeakUp selection and synthesis along with other related topics.