Table of Contents
Inequalities are closely related to equations, but instead of saying two expressions are equal, they compare two quantities and tell you which side is larger, smaller, or possibly equal.
In this chapter, we focus on the general idea of inequalities and how they behave. The detailed step-by-step methods for solving and graphing them will be treated in the child chapters.
What an Inequality Is
An inequality compares two quantities using symbols such as:
- $<$ : “less than”
- $>$ : “greater than”
- $\le$ : “less than or equal to”
- $\ge$ : “greater than or equal to$
Examples of inequalities:
- $3 < 7$
- $x > 2$
- $4y + 1 \le 9$
- $5 - t \ge -3$
Just like equations, inequalities can contain variables. Instead of having one exact solution, an inequality usually has many possible solutions.
For example, in the inequality
$$x > 2,$$
every number greater than $2$ (such as $2.1$, $3$, $10$, $100$) is a solution.
In contrast, an equation like
$$x = 2$$
has exactly one solution: $x = 2$.
Types of Inequality Symbols
It is important to distinguish between strict and inclusive inequalities:
- Strict:
- $a < b$: $a$ is strictly less than $b$; $a$ cannot be equal to $b$.
- $a > b$: $a$ is strictly greater than $b$; $a$ cannot be equal to $b$.
- Inclusive:
- $a \le b$: $a$ is less than or equal to $b$; $a$ may be smaller or exactly equal.
- $a \ge b$: $a$ is greater than or equal to $b$; $a$ may be larger or exactly equal.
This difference matters when you describe solution sets, for example on a number line or using interval notation (introduced elsewhere).
Inequalities vs. Equations
Inequalities and equations have similar structure but different meanings:
- Equation: $2x + 1 = 7$
Says the expression x + 1$ equals $ for certain $x$ values. - Inequality: $2x + 1 < 7$
Says the expression x + 1$ is less than $ for certain $x$ values.
Both can be transformed using similar algebraic moves (adding, subtracting, multiplying, etc.), but inequalities have one special rule that equations do not: multiplying or dividing by a negative number reverses the inequality sign. The child chapter “Solving inequalities” will focus on these methods.
Solutions of Inequalities as Sets
For an equation like $x + 3 = 5$, there is a single solution, $x = 2$.
For an inequality like $x + 3 < 5$, the solution is all numbers less than $2$:
$$x < 2.$$
So the solution to an inequality is often:
- A whole interval of numbers (e.g., all numbers less than $2$),
- Possibly including endpoints (for $\le$ or $\ge$),
- Or excluding endpoints (for $<$ or $>$).
Some examples:
- $x > -1$: all real numbers greater than $-1$.
- $x \le 4$: all real numbers less than or equal to $4$.
- $-2 < x < 3$: all real numbers strictly between $-2$ and $3$.
The way these sets are shown visually on a number line and in coordinate graphs is treated in the “Graphing inequalities” section.
Basic Transformations of Inequalities
You can often manipulate inequalities using similar operations as with equations, provided you keep the rules in mind.
Here are the key ideas, stated without full solving procedures:
- Adding or subtracting the same number
If $a < b$, then: - $a + c < b + c$
- $a - c < b - c$
Example:
From $3 < 5$, adding $2$ to both sides gives $3 + 2 < 5 + 2$, so $5 < 7$.
- Multiplying or dividing by a positive number
If $a < b$ and $c > 0$, then: - $ac < bc$
- $\dfrac{a}{c} < \dfrac{b}{c}$
Example:
From $2 < 4$, multiplying both sides by $3$ gives $6 < 12$.
- Multiplying or dividing by a negative number (sign reversal)
If $a < b$ and $c < 0$, then: - $ac > bc$
- $\dfrac{a}{c} > \dfrac{b}{c}$
The inequality symbol reverses direction.
Example:
From $2 < 4$, multiply both sides by $-1$:
$-2 > -4$ (notice $<$ changed to $>$).
The details of using these rules to isolate a variable and find solution sets belong to the “Solving inequalities” section.
Combining Inequalities
Inequalities can be combined to describe more precise ranges. Two common ways of combining them are:
- And (both conditions must be true):
- Example: $x > 1$ and $x < 4$
Often written as a double inequality:
$ < x < 4.$$ - Or (at least one condition must be true):
- Example: $x \le -2$ or $x \ge 3$
This describes two separate regions on the number line.
The logical ideas behind “and/or” combinations come from basic logic; here they simply describe how inequalities can shape solution sets. How these combinations look when graphed is explored in the “Graphing inequalities” chapter.
Inequalities in Everyday Contexts
Inequalities appear naturally in real situations where conditions involve limits, bounds, or comparisons:
- Age limits: “You must be at least 13 years old” $\rightarrow$ age $\ge 13$.
- Speed limits: “Do not exceed 60 km/h” $\rightarrow$ speed $\le 60$.
- Budgets: “You can spend at most \$50” $\rightarrow$ cost $\le 50$.
- Minimum requirements: “You need more than 70 points to pass” $\rightarrow$ score $> 70$.
Translating word descriptions like these into inequalities is an important skill, especially for word problems. The methods for solving those will rely on the general understanding built in this chapter and the techniques in the child chapters.
Inequalities and the Number Line
A number line gives a simple visual way to show which numbers satisfy an inequality:
- For $x > a$, the solution is all points to the right of $a$.
- For $x < a$, the solution is all points to the left of $a$.
- For $x \ge a$ or $x \le a$, the point $a$ itself is also included.
The specific drawing conventions (open or closed circles, shading directions) and how this connects to graphs on the coordinate plane are covered in the “Graphing inequalities” chapter.
Inequalities as a Step Toward Functions and Graphs
Understanding inequalities is important for later topics:
- Describing domains and ranges (e.g., $x \ge 0$),
- Describing solution regions for equations and systems (e.g., all points $(x,y)$ satisfying $y > 2x + 1$),
- Interpreting constraints in optimization problems.
In this introductory chapter, the main idea is that inequalities compare quantities and usually describe whole sets of solutions, often represented as intervals on a number line or regions in the plane. The next sections will show how to systematically solve them and draw their solution sets.