Writing and Graphing One-Variable Inequalities

Equations are mathematical statements relating two equal quantities. There are also times when it's necessary to relate two quantities that are not equal. For this, inequalities are used.

Concept

Inequality

An inequality, similar to an equation, is a mathematical statement comparing two quantities. Some inequalities indicate that the two quantities are not necessarily equal, while others are strictly never equal.

Symbol	Meaning	Strict / Non-strict
$<$	is less than	Strict
$\leq$	is less than or equal to	Non-strict
$>$	is greater than	Strict
$\geq$	is greater than or equal to	Non-strict

With an inequality, it's possible to compare any combination of two numbers, variables, numeric expressions, or algebraic expressions.

Concept

Solution Set of an Inequality

When an inequality contains an unknown variable, it's possible to solve the inequality. A solution of an inequality is any value of the variable that makes the inequality true. For instance, the inequality $x + 2 < 5$

has the solution

x = 1,

because replacing

x

with

1

yields

3 < 5,

a true statement. Notice that

x = 1

is not the only value that solves

x + 2 < 5;

x = 0

and

x = 2

also work. In fact, most inequalities have an infinite number of solutions. The set of these solutions is called the solution set.

Determine if the number is part of the solution set

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Exercise

Is $r = 2$ an element of the solution set of the following inequality? $\frac{3}{r} \leq - 1$

Show Solution

Solution

If a value is a solution to an inequality, it is an element of the solution set. To determine if $r = 2$ is a solution, we can substitute the value into the inequality and evaluate. If the resulting inequality is true, the value is a solution; if it's false, it is not a solution.

\frac{3}{r} \leq ? - 1

Substitute

r = 2

\frac{3}{2} \leq ? - 1

WriteDecWrite as a decimal

1.5 ≰ - 1

The inequality

1.5 \leq - 1,

read as

‘ ‘ 1.5

is less than or equal to

- 1, "

is a false statement. Thus,

r = 2

is not a solution of the inequality, nor is it an element of the solution set.

Concept

Graphing an Inequality on a Number Line

The graph of a one-variable inequality is a visual representation of the inequality's solution set, which can be drawn on a number line in three steps:

Identify if the inequality is strict.
Draw the boundary point;

use an open point $(\circ)$ for strict or
use a closed point $(∙)$ for non-strict.

Shade the rest of the solution set.

An arrow in either direction indicates that all numbers in that direction are part of the solution set.

x \geq 0

x \leq 2

x > - 3

The inequality

x \geq 0

is graphed with a closed circle at

x = 0

and an arrow pointing to the right. Since

0

solves the inequality, the circle is closed. Additionally, the direction of the arrow points toward numbers greater than

0

, since the inequality symbol is

\geq .

Graph the inequality

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Exercise

Graph the inequality $x < 9$ on a number line.

Show Solution

Solution

The inequality reads $‘ ‘ x$ is less than $9, "$ which means that $x = 9$ is not a solution to the inequality, but every value of $x$ less than $9$ is. Thus, our graph must have an open circle at $x = 9 .$

Every value of $x$ less than $9$ also has to be included in the graph. Since smaller numbers lie to the left on the number line, this is graphed as an arrow pointing to the left.

We have now fully graphed the inequality $x < 9 .$

Write the inequality from its graph

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Exercise

Write the inequality shown by the graph.

Show Solution

Solution

To begin, let's use the variable $x$ in the inequality. From the graph, we can see a closed circle at $- 3 .$ Therefore, $x = - 3$ is a solution of the inequality. The arrow pointing toward the right indicates that all values greater than $3$ are also part of the solution set. Thus, the inequality would read $‘ ‘ x$ is greater than or equal to $- 3, "$ which is expressed algebraically as $x \geq - 3 .$

Writing and Graphing One-Variable Inequalities

Inequality

Solution Set of an Inequality

Example

Determine if the number is part of the solution set

Graphing an Inequality on a Number Line

Example

Graph the inequality

Example

Write the inequality from its graph