# Inequalities

### Writing Inequalities

Algebraic inequalities are used to represent verbal descriptions of problems.

Writing algebraic inequalities is very similar to writing equations. The difference is that instead of setting two expressions equal to each other, the expressions are compared to each other. For example, one side of an inequality may be greater than the other side. This is shown using inequality symbols.

Be sure to read the text carefully to distinguish "less than," which means subtraction, from "is less than," which is an inequality.

• The description "4 less than $x$" is translated as $x-4$.
• The description "4 is less than $x$" is translated as $4\lt{x}$.
Symbol Algebraic Sentence Verbal Sentence
$\lt$ $x\lt 5$ $x$ is less than 5.
$\leq$ $a \le b$ $a$ is less than or equal to $b$.
$\gt$ $9 \gt 8$ 9 is greater than 8.
$\geq$ $500\geq y$ 500 is greater than or equal to $y$.
$\neq$ $3 \neq 5$ 3 is not equal to 5.

Step-By-Step Example
Writing Algebraic Inequalities

Chandra wants to eat less than 1,800 calories each day. Today, she plans to eat 550 calories at breakfast, 700 calories at lunch, and 400 calories at dinner. She also plans to eat some number of crackers that are 10 calories each.

Write an inequality that describes how many calories Chandra plans to eat today.

Step 1

Name the unknown quantity.

Chandra plans to eat some number of crackers.

Let $c$ represent the number of crackers.

Step 2
Write an expression for the number of calories of crackers. The crackers are 10 calories each. So, the number of calories is $10c$.
Step 3

Write an expression for the number of calories Chandra plans to eat today.

Today, she plans to eat 550 calories at breakfast, 700 calories at lunch, 400 calories at dinner, and some number of crackers that are 10 calories each, or $10c$. Add the amounts to get the expression for the number of calories Chandra plans to eat today:
$550+700+400+10c$
Step 4

Identify the parts of the inequality.

Chandra wants to eat less than 1,800 calories each day. So, use the less-than symbol to represent how many calories Chandra wants to eat each day:
$\text{Calories each day} \lt 1,800$
This means that the inequality for the number of calories for today should also use the less-than symbol:
\begin{aligned}\text{Calories for today} &\lt 1,800\\550+700+400+10c &\lt 1,800 \end{aligned}
Step 5
Simplify the inequality.
\begin{aligned}550+700+400+10c &\lt 1,800\\1,650+10c&\lt 1,800\end{aligned}
Solution
The inequality for the number of calories that Chandra plans to eat today is:
$1,650+10c\lt1,800$
The variable $c$ represents the number of crackers.

### Solution Sets of Inequalities

An inequality can have infinitely many solutions. The set of solutions of an inequality can be written using set-builder or interval notation.

Consider the inequality $x\lt10$. There are many values that make this inequality true, such as $x=9$, $x=8$, and so on. Including fractions and negative numbers, there are infinitely many values of $x$ that are solutions. Inequalities often have an infinite number of solutions. This makes it impossible to list all of the solutions. The set of solutions, or solution set, can be described using the inequality $x\lt10$ or by using set-builder notation or interval notation. Another instance is the solution set of $x\gt4.54$. It can be expressed as $\left \{x|x\gt4.54\right \}$, which is read as "the set of all elements $x$, such that $x$ is greater than 4.54."

Interval notation is another way to express the solution set of an inequality. It lists the end points of the solution set separated by a comma. These end points are enclosed by parentheses, brackets, or a combination of the two to indicate whether the values are included in the solution set. If the symbol is exclusive ($\lt, \gt$), then parentheses are used, and the interval is called an open interval. If the symbol is inclusive ($\leq, \geq$), then brackets are used, and the interval is called a closed interval. If a combination of symbols is used, then use one of each on the appropriate side; the interval is called a half-open interval.

Therefore, one solution set can be expressed in several different, equivalent ways. Note that infinite solution sets are expressed in interval notation using the infinity symbol, $\infty$. Parentheses are used instead of brackets because infinity is not a number. So it cannot be part of the solution set.

### Set-Builder Notation

 $\lbrace$ $x$ $\vert$ $\rbrace$ the set of all elements $x$ such that $x$ meets certain conditions

### Grouping Symbols and End Points for Inequalities

 Inequality Symbol $\lt$ $\gt$ $\leq$ $\geq$ Associated Grouping Symbol ( ) ( ) [ ] [ ] Number Line End Point ∘ ∘ • •

### Examples of Solution Sets

Interval Notation Number Line Set-Builder Notation
$(3, 6]$
$\left \{x|3\lt{x}\lt{6} \right \}$

The set of all elements $x$ such that 3 is less than $x$ and $x$ is less than 6
$(3, 6]$
$\left \{x|3\lt{x}\leq6 \right \}$

The set of all elements $x$ such that 3 is less than $x$ and $x$ is less than or equal to 6
$(3, \infty)$
$\left \{x|3\lt{x} \right \}$

The set of all elements $x$ such that 3 is less than $x$
$[-5, 4]$
$\left \lbrace x\vert-\! 5\leq x\leq 4 \right\rbrace$

The set of all elements $x$ such that –5 is less than or equal to $x$ and $x$ is less than or equal to 4
$[-5, 4)$
$\left \{x|-\!5\leq x\lt 4\right \}$

The set of all elements $x$ such that –5 is less than or equal to $x$ and $x$ is less than 4
$(-\infty, 4]$
$\left \{x|x\leq4 \right \}$

The set of all elements $x$ such that $x$ is less than or equal to 4
$(-\infty, \infty)$
$\left \{x|x\;\text{is a real number}\right \}$

The set of all real numbers, which include whole numbers, negative numbers, positive numbers, fractions, and decimals

Solution sets of inequalities can be represented in three different ways. They include set-builder notation, a number line, and interval notation.

### Checking Solutions of Inequalities

To check a solution of an inequality, substitute the value of the variables in the inequality, and determine whether the resulting statement is true.
An inequality is true if the comparison is true. Solutions of an inequality make the inequality true. For example, $x=0$ and $x=1$ are both solutions of the inequality:
$x+2\lt 4$
However, $x=2$ and $x=3$ are not solutions to the inequality. The process of checking a number to see whether it is a solution of an inequality is almost the same as checking solutions of equations. Substitute the value into the inequality and evaluate both sides. If the resulting inequality is true, then it is a solution.