Skip to content# Inequalities on a Number Line: Examples with Practice Questions

## Inequalities on a Number Line

## Understanding Inequality Symbols

## Representing Inequalities on a Number Line

## Including the Endpoint

## Summary of Symbols and Circles

## Compound Inequalities

## Important Notes on Inequality Direction

## Practice Problems

## Conclusion

## Practice Questions and Answers on Inequalities on a Number Line

## Solutions

In this article, we will discuss how to solve inequalities and represent them on a number line.

**Video Tutorial on GCSE Maths: Inequalities on a Number Line**

Watch this Video Tutorial as we explain all types of Inequalities on a Number Line for GCSE Maths.

Inequalities are fundamental in algebra and help us understand the range of possible values that satisfy a condition.

We will discuss are:

**The Basic Inequality Symbols****How to plot them on a number line****Work through examples****Including compound inequalities.**

*They are very important in practicing questions for coordinate geometry as well.*

Here is one more link to practice a few extra questions: Maths Genie Inequalities on a Number Line Questions

Inequalities express a relationship where two values are not equal and one is greater or lesser than the other. The primary inequality symbols are:

**Less than (<):**Indicates that one value is smaller than another.**Greater than (>):**Indicates that one value is larger than another.**Less than or equal to (≤):**Indicates that one value is smaller than or equal to another.**Greater than or equal to (≥):**Indicates that one value is larger than or equal to another.

A number line is a visual tool that helps illustrate the set of values that satisfy an inequality.

**Key Steps to Plot an Inequality:**

**Draw a Number Line:**Sketch a horizontal line and mark relevant numbers.**Identify Key Points:**Mark the number(s) involved in the inequality.**Use Circles to Indicate Inclusion:**

**Open Circle:**Used when the number is not included in the solution (for < or >).**Closed Circle:**Used when the number is included in the solution (for ≤ or ≥).

**4. Shade the Solution Area: **

- Draw an arrow or line extending left or right to represent all possible values that satisfy the inequality.

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**Solved Example 1**

**Question: x<5**

**Solution: **

**Interpretation:**

- x can be any number less than 5 (e.g., 4, 0, -3, 4.9).

**Steps to Plot:**

**Step 1: Draw a number line** and label key points, including 5.

**Step 2: Place an open circle at 5** because 5 is not included.

**Step 3: Shade the line to the left of 5**, indicating all numbers less than 5.

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**Solved Example 2**

**Question: x>5**

**Solution: **

**Interpretation:**

- x can be any number greater than 5 (e.g., 6, 7, 5.1).

**Steps to Plot:**

**Step 1: Draw a number line **and mark the point 5.

**Step 2: Place an open circle at 5.**

**Step 3: Shade the line to the right of 5, **showing all numbers greater than 5.

- When the inequality includes equality (≤ or ≥), the endpoint is part of the solution set.

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**Solved Example 3:**

**Question: x ≤ 5**

**Solution: **

**Interpretation:**

- x can be 5 or any number less than 5.

**Steps to Plot:**

**Step 1: Draw a number line **and label 5.

**Step 2: Place a closed circle at 5 **to include it in the solution.

**Step 3: Shade the line to the left of 5.**

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**Solved Example 4: **

**Question: x ≥ 5**

**Solution: **

**Interpretation:**

- x can be 5 or any number greater than 5.

**Steps to Plot:**

**Step 1: Draw a number line **and label 5.

**Step 2: Place a closed circle at 5.**

**Step 3: Shade the line to the right of 5.**

**Open Circle:**Used for < and > (number not included).**Closed Circle:**Used for ≤ and ≥ (number included).

- Compound inequalities involve two inequality symbols and define a range between two values.

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**Solved Example 5: **

**Question: 1 < x ≤ 5**

**Solution: **

**Interpretation:**

- x is greater than 1 but less than or equal to 5.

**Steps to Plot:**

**Step 1: Draw a number line **and label points 1 and 5.

**Step 2: Place an open circle at 1 **(since x is not equal to 1).

**Step 3: Place a closed circle at 5** (since x can be equal to 5).

**Step 4: Shade the region between 1 and 5, **connecting the two circles.

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**Solved Example 6: **

**Question: −2 ≤ x < 3**

**Solution: **

**Interpretation:**

- x is greater than or equal to -2 but less than 3.

**Steps to Plot:**

**Step 1: Draw a number line **and mark -2 and 3.

**Step 2: Place a closed circle at -2 **(including -2 in the solution)

**Step 3: Place an open circle at 3 **(excluding 3)

**Step 4: Shade the region between -2 and 3.**

**Direction Matters:**The inequality symbol points towards the smaller value**Equivalent Expressions:**- x>2 is the same as 2<x
- Both indicate that x is greater than 2.

**Example:**

**2<x**Reads as “2 is less than x,” meaning x is greater than 2.**x>2:**Reads as “x is greater than 2.”

Understanding this helps avoid confusion when interpreting or rearranging inequalities.

Try plotting the following inequalities on a number line:

**1. x ≥ −4**

**Interpretation:**x is -4 or any number greater.**Plot:**Closed circle at -4, shade to the right.

**2. x < 0**

**Interpretation:**x is any number less than 0.**Plot:**Open circle at 0, shade to the left.

**3. −3 < x ≤ 2**

**Interpretation:**x is greater than -3 and up to 2, including 2.**Plot:**Open circle at -3, closed circle at 2, shade between them.

Understanding how to solve inequalities and represent them on a number line is a crucial skill in algebra. Remember:

**Use open circles**for < and > (number not included).**Use closed circles**for ≤ and ≥ (number included).**Shade appropriately**to represent all possible values of x that satisfy the inequality.**Pay attention to the direction**of the inequality symbol.

**Question 1: **Represent the inequality x < 4 on a number line.

**Question 2: **Sketch the inequality x ≥ −3 on a number line.

**Question 3: **Plot the inequality x > 0 on a number line.

**Question 4: **Illustrate the inequality x ≤ 6 on a number line.

**Question 5: **Show the solution of the compound inequality −2 < x ≤ 5 on a number line.

**Question 6: **Represent the inequality x ≥ −7 on a number line.

**Question 7:** Graph the inequality x ≤ 2 on a number line.

**Question 8: **Plot the compound inequality 1 ≤ x < 4 on a number line.

**Question 9:**Illustrate the inequality x > −5 on a number line.

**Question 10: **Show the solution set for the compound inequality −3 ≤ x ≤ 3 on a number line.

**Question 1: **

**Step 1:** **Draw a Horizontal Number Line:**

- Sketch a straight horizontal line.
- Mark evenly spaced intervals.

**Step 2: Mark the Key Point (4):**

- Locate and label the point corresponding to x = 4 on the number line.

**Step 3: Place an Open Circle at 4:**

- Draw an open (hollow) circle at the point labelled 4.
**Reason:**The inequality is “less than” (<), so 4 is not included in the solution set.

**Step 4: Shade to the Left of 4:**

- Draw a line or arrow extending from the open circle to the left.
**Reason:**To represent all real numbers less than 4.

**Question 2:**

**Step 1: Draw a Horizontal Number Line:**

- Sketch a straight horizontal line with intervals.

**Step 2: Mark the Key Point (-3): **

- Locate and label the point corresponding to x = −3.

**Step 3: Place a Closed Circle at -3: **

- Draw a closed (filled-in) circle at -3.
**Reason:**The inequality is “greater than or equal to” (≥), so -3 is included in the solution set.

**Step 4: Shade to the Right of -3: **

- Draw a line or arrow extending from the closed circle to the right.
**Reason:**To represent all real numbers greater than or equal to -3.

**Question 3:**

**Step 1: Draw a Horizontal Number Line: **

- Sketch the number line with appropriate intervals.

**Step 2: Mark the Key Point (0): **

- Locate and label the point x = 0.

**Step 3: Place an Open Circle at 0:**

- Draw an open circle at 0.
**Reason:**The inequality is “greater than” (>), so 0 is not included.

**Step 4: Shade to the Right of 0:**

- Extend a line or arrow from the open circle to the right.
**Reason:**To represent all real numbers greater than 0.

**Question 4:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Point (6): **

- Locate and label x = 6.

**Step 3: Place a Closed Circle at 6: **

- Draw a closed circle at 6.
**Reason:**The inequality is “less than or equal to” (≤), so 6 is included.

**Step 4: Shade to the Left of 6: **

- Draw a line or arrow extending left from the closed circle.
**Reason:**To represent all real numbers less than or equal to 6.

**Question 5:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Points (-2 and 5): **

- Label x = −2 and x = 5.

**Step 3: Place an Open Circle at -2: **

- Draw an open circle at -2.
**Reason:**The inequality is “greater than” (>), so -2 is not included.

**Step 4: Place a Closed Circle at 5: **

- Draw a closed circle at 5.
**Reason:**The inequality is “less than or equal to” (≤), so 5 is included.

**Step 5: Shade the Region Between -2 and 5:**

- Draw a line connecting the two circles.
**Reason:**To represent all real numbers greater than -2 and less than or equal to 5.

**Question 6:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Point (-7): **

- Label x = −7.

**Step 3: Place a Closed Circle at -7: **

- Draw a closed circle at -7.
**Reason:**The inequality is “greater than or equal to” (≥), so -7 is included.

**Step 4: Shade to the Right of -7: **

- Extend a line or arrow from the closed circle to the right.
**Reason:**To represent all real numbers greater than or equal to -7.

**Question 7:**

**Step 1: ****Draw a Horizontal Number Line**.

**Step 2: Mark the Key Point (2): **

- Label x = 2.

**Step 3: Place a Closed Circle at 2: **

- Draw a closed circle at 2.
**Reason:**The inequality is “less than or equal to” (≤), so 2 is included.

**Step 4: Shade to the Left of 2: **

- Draw a line or arrow extending left from the closed circle.
**Reason:**To represent all real numbers less than or equal to 2.

**Question 8:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Points (1 and 4): **

- Label x = 1 and x = 4.

**Step 3: Place a Closed Circle at 1: **

- Draw a closed circle at 1.
**Reason:**The inequality is “greater than or equal to” (≥), so 1 is included.

**Step 4: Place an Open Circle at 4: **

- Draw an open circle at 4.
**Reason:**The inequality is “less than” (<), so 4 is not included.

**Step 5: Shade the Region Between 1 and 4: **

- Draw a line connecting the two circles.
**Reason:**To represent all real numbers from 1 up to (but not including) 4.

**Question 9:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Point (-5): **

- Label x = −5.

**Step 3: Place an Open Circle at -5: **

- Draw an open circle at -5.
**Reason:**The inequality is “greater than” ( > >), so -5 is not included.

**Step 4: Shade to the Right of -5: **

- Extend a line or arrow from the open circle to the right.
**Reason:**To represent all real numbers greater than -5.

**Question 10:**

**Step 1: Draw a Horizontal Number Line.**

**Step 2: Mark the Key Points (-3 and 3): **

- Label x = −3 and x = 3.

**Step 3: Place a Closed Circle at -3: **

- Draw a closed circle at -3.
**Reason:**The inequality is “greater than or equal to” (≥), so -3 is included.

**Step 4: Place a Closed Circle at 3: **

- Draw a closed circle at 3.
**Reason:**The inequality is “less than or equal to” ( ≤ ≤), so 3 is included.

**Step 5: ****Shade the Region Between -3 and 3**:

- Draw a line connecting the two closed circles.
**Reason**: To represent all real numbers between -3 and 3, including both endpoints.

**Table of Content**

- Video Tutorial on GCSE Maths: Inequalities on a Number Line
- Inequalities on a Number Line
- Understanding Inequality Symbols
- Representing Inequalities on a Number Line
- Compound Inequalities
- Important Notes on Inequality Direction
- Practice Problems
- Conclusion
- Practice Questions and Answers on Inequalities on a Number Line