The keyword number linen is interpreted as number line, the standard visual tool used by a Linear Inequality Solver to display the solution set. A number line graph shows which real numbers satisfy an inequality by marking a boundary point and shading the appropriate direction.
Rule 1 Use an open circle at the boundary if the inequality is strict: \(x<a\) or \(x>a\).
Rule 2 Use a closed circle at the boundary if the inequality is inclusive: \(x\le a\) or \(x\ge a\).
Rule 3 Shade left for \(x<a\) or \(x\le a\), and shade right for \(x>a\) or \(x\ge a\).
How a linear inequality becomes a number line graph
A typical linear inequality solver first isolates \(x\) and then translates the result into a number line picture. For example, consider the inequality \(2x-3\le 5\).
Step 1: Solve the inequality algebraically
\[ 2x-3\le 5 \]
\[ 2x\le 8 \]
\[ x\le 4 \]
Step 2: Interpret \(x\le 4\) on the number line
- The boundary value is \(4\).
- The symbol \(\le\) is inclusive, so the boundary point is included: use a closed circle at \(4\).
- All solutions are less than or equal to \(4\): shade to the left.
Solution set notation
The same solution can be written in two standard algebra formats:
| Notation | How it looks for \(x\le 4\) | Meaning |
|---|---|---|
| Interval notation | \((-\infty,4]\) | All real numbers up to and including \(4\) |
| Set-builder notation | \(\{x\in\mathbb{R}: x\le 4\}\) | All real \(x\) that satisfy the inequality |
Visualization: number line for a linear inequality
The diagram shows the solution to \(x\le 4\): a closed dot at \(4\) and shading to the left, indicating that every value less than \(4\) (and also \(4\) itself) satisfies the inequality.
Common pitfalls when using a number line
Mistake Using an open circle for \(\le\) or \(\ge\), or a closed circle for \(<\) or \(>\).
Fix Strict inequalities exclude the boundary (open circle); inclusive inequalities include it (closed circle).
Mistake Shading the wrong direction.
Fix Think “less than” means left, “greater than” means right; test a value (such as \(0\)) if unsure.
Summary
The number linen (number line) method in a Linear Inequality Solver converts an algebraic result like \(x\le 4\) into a visual solution set: mark the boundary with a closed circle for \(\le\) or \(\ge\) (open for \(<\) or \(>\)) and shade the side of the number line that satisfies the inequality.