Algebra 1 · Unit 3 · Topic 3.1

Inequality Symbols and Number-Line Graphs

Read <, >, ≤, and ≥ as solution sets; use open or filled endpoints, rays, and interval notation to graph inequalities on a number line.

One statement can describe many numbers

An equation such as \(x=2\) selects one real number. The inequality \(x>2\) selects every real number larger than \(2\): \(3\) works, \(2.1\) works, and so does a number arbitrarily close to \(2\) from the right. But \(2\) itself does not work. Throughout this lesson \(x\) ranges over the real numbers and has no physical unit. A number-line graph displays that whole solution set, not just a few sample answers.

On a number line, values increase from left to right. The symbols \(<\) and \(>\) are strict: equality at the boundary is excluded. The symbols \(\le\) and \(\ge\) add the possibility of equality. For example, \(x\ge2\) includes \(2\) and everything to its right, whereas \(x\le2\) includes \(2\) and everything to its left. The word “or” is built into “less than or equal to”; this does not mean you must choose only one of those cases.

Mark the boundary before drawing a ray

First locate the boundary number. Then decide whether it belongs to the solution set by substituting that number into the inequality. Use an open circle if it fails the inequality and a filled circle if it satisfies it. Finally, draw a ray toward the numbers that work. The direction follows the values, not the shape or pointing direction of the printed inequality symbol.

The original diagram below compares all four symbols using the same boundary \(2\). Keeping the boundary fixed makes the two separate choices visible: left versus right, and excluded versus included. The teal portion is the solution set; the gray line shows the surrounding real numbers.

Four original number-line graphs with boundary 2. For x greater than 2, an open circle at 2 has a ray to the right. For x greater than or equal to 2, a filled circle at 2 has a ray to the right. For x less than 2, an open circle at 2 has a ray to the left. For x less than or equal to 2, a filled circle at 2 has a ray to the left. Every line is labeled from negative 4 through 4. — Open means the boundary is out; filled means it is in. Arrows show that the solution set continues beyond the displayed number-line window.

For the top two rows, \(3>2\) and \(3\ge2\) are both true. The difference appears at \(2\): \(2>2\) is false, but \(2\ge2\) is true. The bottom two rows make the same distinction on the left: \(1<2\) and \(1\le2\) are true, while only \(2\le2\) includes the boundary.

Read the inequality and its graph in either direction

Consider the existing example \(x\ge-2\). Test the endpoint: \(-2\ge-2\) is true, so place a filled point at \(-2\). Since \(-1\ge-2\) and \(0\ge-2\), continue the ray to the right. But \(-3\ge-2\) is false, so the left side is not shaded. One endpoint test and one nearby test are enough to determine how to draw this simple ray; they do not replace the meaning of “all numbers greater than or equal to \(-2\).”

The interval notation for the same real-number set is \(\left[-2,\infty\right)\). The square bracket includes \(-2\). The parenthesis next to \(\infty\) is always used because infinity is not a real number that can be reached or included. The ray's arrow also means “continues without bound,” not “includes infinity.”

As a contrast, \(x<3\) has an open circle at \(3\), shades left, and is written \(\left(-\infty,3\right)\). In either notation, the boundary decision is the same: \(3<3\) is false. If a statement is written as \(3>x\), it says exactly the same thing as \(x<3\); rewriting it with \(x\) first changes the orientation of the written symbol because the two sides have swapped places, not because any algebraic operation was performed.

Inequality with boundary \(a\)	Endpoint at \(a\)	Direction of solution ray	Interval notation
\(x>a\)	Open	Right	\(\left(a,\infty\right)\)
\(x\ge a\)	Filled	Right	\(\left[a,\infty\right)\)
\(x<a\)	Open	Left	\(\left(-\infty,a\right)\)
\(x\le a\)	Filled	Left	\(\left(-\infty,a\right]\)

This continuous shading assumes \(x\) can be any real number. If a question instead says that \(n\) counts whole objects, then \(n>2\) allows \(3,4,5,\ldots\), not every real number between them. The inequality symbol has the same meaning, but the stated domain changes which points can be solutions.

When checking a number-line graph, ask three separate questions: Is the boundary at the correct number? Does the endpoint's fill agree with equality? Does the ray point toward values that actually satisfy the statement? This is more reliable than trying to memorize an arrow direction from the visual shape of \(<\) or \(>\).