Table of Contents
In this chapter we focus on how to represent inequalities on a number line. Solving inequalities is handled in the parent chapter; here we assume you already have an inequality (like $x > 3$) and want to graph its solutions.
The Number Line as a Picture of Solutions
An inequality in one variable (like $x$) usually has many solutions, not just one number. A graph on the number line is a way to show all those solutions at once.
We will use:
- A horizontal line with tick marks for numbers.
- A special mark at the “boundary” number (like $3$ in $x > 3$).
- Shading (or an arrow) to show which side of the boundary is included.
Open and Closed Points
The first decision when graphing an inequality is whether the boundary number itself is a solution.
- If the inequality includes the boundary number:
- Use a closed (filled-in) dot.
- If the inequality does not include the boundary number:
- Use an open (hollow) dot.
This matches the inequality symbols:
- $x > a$ means “strictly greater than $a$”:
- $a$ is not a solution → open dot at $a$.
- $x < a$ means “strictly less than $a$”:
- $a$ is not a solution → open dot at $a$.
- $x \ge a$ means “greater than or equal to $a$”:
- $a$ is a solution → closed dot at $a$.
- $x \le a$ means “less than or equal to $a$”:
- $a$ is a solution → closed dot at $a$.
Direction of Shading
After marking the boundary with an open or closed dot, you show all other solutions by shading to the left or right.
- For $x > a$ or $x \ge a$:
- Shade to the right of $a$.
- For $x < a$ or $x \le a$:
- Shade to the left of $a$.
The shading usually extends with an arrow to show it goes on forever in that direction.
Examples
- Graph $x > 3$
- Boundary: $3$.
- Symbol is $>$, so $3$ is not included: open dot at $3$.
- “Greater than” → shade to the right.
You would draw: an open circle at $3$, shading and an arrow going to the right.
- Graph $x \le -1$
- Boundary: $-1$.
- Symbol is $\le$, so $-1$ is included: closed dot at $-1$.
- “Less than or equal” → shade to the left.
You would draw: a solid dot at $-1$, shading and an arrow going to the left.
Graphing Inequalities with “At Least” and “At Most”
Words often used in problems correspond to inequality symbols and influence the graph:
- “At least $a$” → $x \ge a$:
- Closed dot at $a$, shade right.
- “More than $a$” → $x > a$:
- Open dot at $a$, shade right.
- “At most $a$” → $x \le a$:
- Closed dot at $a$, shade left.
- “Less than $a$” → $x < a$:
- Open dot at $a$, shade left.
For example, “the temperature is at least $5^\circ$C” means $T \ge 5$:
closed dot at $5$, shading to the right.
Graphing Compound Inequalities on a Number Line
Sometimes you see inequalities that limit a variable between two numbers, such as:
$$
2 < x \le 7
$$
This means “$x$ is greater than $2$ and less than or equal to $7$.”
To graph this:
- Mark the two boundary points:
- At $2$: symbol is $<$, so use an open dot.
- At $7$: symbol is $\le$, so use a closed dot.
- Shade between $2$ and $7$ (because $x$ must satisfy both conditions).
- Do not shade outside that interval, since values smaller than $2$ or larger than $7$ are not solutions.
Another example:
$$
x \ge -4 \quad \text{and} \quad x < 1
$$
can be written as
$$
-4 \le x < 1
$$
and is graphed with:
- Closed dot at $-4$.
- Open dot at $1$.
- Shading in between $-4$ and $1$ only.
Inequalities with “Or”
Some compound inequalities use “or,” such as:
$$
x < -2 \quad \text{or} \quad x \ge 3.
$$
This means $x$ can be in either region:
- For $x < -2$:
- Open dot at $-2$, shade to the left.
- For $x \ge 3$:
- Closed dot at $3$, shade to the right.
On the number line, you will see two separate shaded parts, with nothing shaded between $-2$ and $3$.
Checking a Graph Against an Inequality
To avoid mistakes, you can test your graph:
- Pick a number from the shaded region.
- Substitute it into the inequality and check if it makes a true statement.
- Pick a number from the unshaded region.
- Substitute and check that it makes a false statement.
If both tests work, your graph likely matches the inequality correctly.
Interval Notation and Graphs (Preview)
You may later see interval notation, which is another way to describe the same sets of solutions:
- Open dot (endpoint not included) corresponds to a parenthesis: $(a$ or $a)$.
- Closed dot (endpoint included) corresponds to a bracket: $[a$ or $a]$.
- Arrows on the number line correspond to extending to $-\infty$ or $+\infty$.
For example:
- $x > 3$:
- Graph: open dot at $3$, shade to the right.
- Interval notation: $(3, \infty)$.
- $2 \le x < 5$:
- Graph: closed at $2$, open at $5$, shade between.
- Interval notation: $[2, 5)$.
The detailed rules for interval notation can be explored further when that topic is formally introduced; here it is enough to see how it matches your graphs.
Summary of Steps for Graphing a One-Variable Inequality
- Identify the boundary number(s).
- Decide open vs. closed dot:
- $<$ or $>$ → open.
- $\le$ or $\ge$ → closed.
- Decide where to shade:
- $>$ or $\ge$ → right.
- $<$ or $\le$ → left.
- For compound inequalities:
- “And” → overlapping region (often a single segment).
- “Or” → union of separate regions.
- Optionally test a number from shaded and unshaded parts to confirm.