4.2.1 An Introduction to Inequalities

Learning Outcomes

  • Represent inequalities on a number line
  • Represent inequalities using interval notation
  • Describe solutions to inequalities

Key WORDS

  • Inequality: a mathematical statement that compares two expressions using the ideas of greater than or less than
  • Real number line: a line that represents all real numbers from negative infinity to infinity
  • Interval notation: sets with parentheses representing the strict inequalities < or > and or brackets representing the inequalities ≤ or ≥
  • Solution set: the set formed by all solutions of an inequality
  • Set-builder notation: defines the variable and uses an inequality inside a pair of braces

 

Inequalities

An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. We use the following symbols with inequalities.

Words Symbol Examples
Less than < 3 < 5     -7 < -4     0 < 1
Less than or equal to ≤ 6 ≤ 9     -9 ≤ 0      5 ≤ 5
Greater than > 0 > 5     -3 > -7     9 > -1
Greater than or equal to ≥ 8 ≥ 8       7 ≥ 4    -2 ≥ -3

When we have inequalities that involve variables, there is often an infinite number of values the variable can take.

For example:

  • [latex]{x}\lt{9}[/latex] indicates the list of numbers that are less than [latex]9[/latex].
  • [latex]-5\le{t}[/latex] indicates all the numbers that are greater than or equal to [latex]-5[/latex].

Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.

For example:

  • [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.

Representing Inequalities on a Number Line

As we saw in 1.1.2, inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs.  Graphs are a very helpful way to visualize information, especially when that information represents an infinite list of numbers.

[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 point is part of the solution. An open circle is used for greater than (>) or less than (<). The point is not part of the solution.

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]

Solution

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.

Try It

 

Example

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

Solution

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.

Interval Notation

Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called interval notation. We introduced this convention in 1.1.2. when we built sets with parentheses representing the strict inequalities < or > and or brackets representing the inequalities ≤ or ≥. We also 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.” For example, [latex]x \in [6,\,\infty)[/latex] represents all x-values that are greater than or equal to 6.

A few examples of an interval, or a set of numbers in which a solution falls, are [latex]\left[-2,6\right)[/latex], or all numbers between [latex]-2[/latex] and [latex]6[/latex], including [latex]-2[/latex], but not including [latex]6[/latex]; [latex]\left(-1,0\right)[/latex], all real numbers between, but not including [latex]-1[/latex] and [latex]0[/latex]; and [latex]\left(-\infty,1\right][/latex], all real numbers less than and including [latex]1[/latex]. The table below outlines the possibilities. Remember to read inequalities from left to right, just like text.

The table below describes all the possible inequalities that can occur and how to write them using interval notation, where a and b are real numbers.

Inequality Words Interval Notation
[latex]{a}\lt{x}\lt{ b}[/latex] all real numbers between a and b, not including a and b [latex]\left(a,b\right)[/latex]
[latex]{x}\gt{a}[/latex] All real numbers greater than a, but not including a [latex]\left(a,\infty \right)[/latex]
[latex]{x}\lt{b}[/latex] All real numbers less than b, but not including b [latex]\left(-\infty ,b\right)[/latex]
[latex]{x}\ge{a}[/latex] All real numbers greater than a, including a [latex]\left[a,\infty \right)[/latex]
[latex]{x}\le{b}[/latex] All real numbers less than b, including b [latex]\left(-\infty ,b\right][/latex]
[latex]{a}\le{x}\lt{ b}[/latex] All real numbers between a and b, including a [latex]\left[a,b\right)[/latex]
[latex]{a}\lt{x}\le{ b}[/latex] All real numbers between a and b, including b [latex]\left(a,b\right][/latex]
[latex]{a}\le{x}\le{ b}[/latex] All real numbers between a and b, including a and b [latex]\left[a,b\right][/latex]
[latex]{x}\lt{a}\text{ or }{x}\gt{ b}[/latex] All real numbers less than a or greater than b [latex]\left(-\infty ,a\right)\cup \left(b,\infty \right)[/latex]
All real numbers All real numbers [latex]\left(-\infty ,\infty \right)[/latex]

Example

Describe the inequality [latex]x\ge 4[/latex] using interval notation
Solution

The solutions to [latex]x\ge 4[/latex] are represented as [latex]\left[4,\infty \right)[/latex].

Note the use of a bracket on the left because 4 is included in the solution set.

Try It

Example

Use interval notation to indicate all real numbers greater than or equal to [latex]-2[/latex].

Solution

Use a bracket on the left of [latex]-2[/latex] and parentheses after infinity: [latex]\left[-2,\infty \right)[/latex]. The bracket indicates that [latex]-2[/latex] is included in the set with all real numbers greater than [latex]-2[/latex] to infinity.

 

The following video shows examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.

Set-Builder Notation

The solutions to inequalities form a set of numbers, so set-builder notation (see section 1.1.2) can also be used to represent the solution set. Set-builder notation defines the variable and uses an inequality inside a pair of braces. For example, the solution set [latex]\left ( 4,\,\infty \right )[/latex] is written in set-builder notation as [latex]\{ x\;\large |\;\normalsize x\gt 4,\;x\in\mathbb{R} \; \}[/latex]. This is read the set of all [latex]x[/latex]-values such that [latex]x[/latex] is a real number greater than 4.

Example

Convert the solution set from interval notation to set-builder notation:  [latex]x\in \left (-\infty,\; 7\right ][/latex]

Solution

[latex]x[/latex] lies between [latex]-\infty[/latex] and [latex]7[/latex], so [latex]x[/latex] is less than or equal to [latex]7[/latex].

[latex]\{\;x\;\large|\;\;x\le 7,\;\;x\in\mathbb{R}\}[/latex]

Example

Convert the solution set from set-builder notation to interval notation: [latex]\{\;x\;\large|\;\;x\ge -2,\;\;x\in\mathbb{R}\}[/latex]

Solution

[latex]x[/latex] is greater than or equal to [latex]-2[/latex] so the interval starts at [latex]-2[/latex] and goes to [latex]\infty[/latex].

[latex]x\in\left [ -2,\;\infty\right )[/latex]

Try It

1. Convert the solution set from interval notation to set-builder notation:  [latex]x\in \left (-3,\;\infty\right )[/latex]

 

2. Convert the solution set from set-builder notation to interval notation: [latex]\{\;x\;\large|\;\;x\le 8,\;\;x\in\mathbb{R}\}[/latex]