Linear Inequalities: Number Line

Table of Contents

1. Why Use a Number Line?
2. Open and Closed Circles
3. Representing Basic Inequalities
4. Reading Solutions from a Number Line
5. From Inequality to Number Line
6. Common Mistakes
7. Practice Exercises
8. Related Articles

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1. Why Use a Number Line?

A number line gives you a visual picture of all the numbers that satisfy an inequality. Instead of just writing $x>3$, you can see at a glance which part of the number line is included in the solution.

Key idea: A number line diagram shows the boundary point and the direction of the solution set using circles and shaded regions.

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2. Open and Closed Circles

There are two types of circles used on the number line:

Remember: Strict inequalities ($<$, $>$) use an open circle. Non-strict inequalities ($\le$, $\ge$) use a filled circle.

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3. Representing Basic Inequalities

Example 1: $x>3$ (strict, greater than)

The solution includes all numbers to the right of 3, but not 3 itself.

• Open circle at 3 (3 is not included)
• Blue region extends to the right toward $+\infty$
• Interval notation: $(3,\infty )$

Example 2: $x\le -2$ (non-strict, less than or equal)

The solution includes all numbers to the left of $-2$, including $-2$ itself.

• Filled circle at $-2$ ($-2$ is included)
• Blue region extends to the left toward $-\infty$
• Interval notation: $(-\infty ,-2]$

Example 3: $x\ge 0$ (non-strict, greater than or equal)

The solution includes zero and all positive numbers.

• Filled circle at 0 (0 is included)
• Blue region extends to the right
• Interval notation: $[0,\infty )$

Example 4: $x<5$ (strict, less than)

The solution includes all numbers to the left of 5, but not 5 itself.

• Open circle at 5 (5 is not included)
• Blue region extends to the left toward $-\infty$
• Interval notation: $(-\infty ,5)$

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4. Reading Solutions from a Number Line

When you see a number line diagram, you can convert it back to an inequality:

1. Identify the boundary point (the number under the circle)
2. Check the circle type --- open means strict, filled means non-strict
3. Check the direction of the shaded region

Tip: Think of the open circle as a "hole" --- the boundary value has been removed from the solution.

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5. From Inequality to Number Line

Here is a step-by-step process for drawing any linear inequality on a number line:

Step 1: Solve the inequality to get it in the form $x>a$, $x<a$, $x\ge a$, or $x\le a$. Step 2: Draw a horizontal line and mark the boundary value $a$. Step 3: Draw the correct circle at $a$:

• Open circle for $<$ or $>$
• Filled circle for $\le$ or $\ge$

Step 4: Shade the correct direction:

• Shade to the right for $>$ or $\ge$
• Shade to the left for $<$ or $\le$

Worked Example

Solve and represent on a number line: $3x-6>9$

Solving: Number line: Open circle at 5, shade to the right. Solution: $x\in (5,\infty )$

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6. Common Mistakes

Mistake 1: Mixing Up Circle Types

For $x>3$: use an open circle (3 is not a solution).

For $x\ge 3$: use a filled circle (3 is a solution).

Mistake 2: Shading the Wrong Direction

For $x<5$: shade to the left (toward smaller numbers).

For $x>5$: shade to the right (toward larger numbers).

Mistake 3: Forgetting to Solve First

Always solve the inequality completely before drawing the number line. Do not try to graph $3x-6>9$ directly --- first simplify to $x>5$.

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Practice Exercises

• Inequalities - Number Line - Draw and read number line diagrams
• Inequalities - Intervals - Convert between number lines and intervals
• Inequalities - Basic - Solve and graph simple inequalities

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Related Articles

• Linear Inequalities - Comprehensive Guide - Full introduction to linear inequalities
• Linear Inequalities - Intervals and Sets - Writing solutions with interval notation
• Linear Inequalities - Simple Inequalities - Solving basic inequality types
• Linear Inequalities - Rules and Formulas - Quick reference card