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Inequalities - Representation on the Number Line

We'll often need to find all of the numbers that satisfy a given inequality. This is known as solving an inequality
For instance we may be asked to represent all of the numbers that satisfy the inequality: \[x < 5 \] In other words we need to state all the numbers for which that inequality is true. There are an infinite amount of numbers for which this inequality is true. Indeed, so long as \(x\) is a number that is less than \(5\) this inequality will be true. So \(4.9,\ 4,\ 3,\ 1,\ 0,\ -7,\ -52, \ \dots \) are all numbers that satisfy this inequality ("solutions to the inequality").

Since it isn't possible to write all of the numbers that satisfy this inequality, we often use a graphical representation of the solution. We do so using the number line.

The representation of the solutions of the inequality \(x<5\) is shown here:

The pink arrow starts at \(5\) and illustrates all of the numbers that satisfy the inequality.
The arrow points in the direction of all of the solutions.

Method - Inequalities on the Number Line

Each of the four inequality symbols is illustrated here.
These

  • \(x > a \) : \(x\) is greater than \(a\).
    This is represented by an arrow with an empty dot above the \(a\).
    The arrow points in the direction of all the numbers that are greater than \(a\).
    The empty dot highlights that fact that \(x\) cannot equal \(a\):


  • \(x \geq a \) : \(x\) is greater than or equal to \(a\).
    This is represented by an arrow with an filled-in dot above the \(a\).
    The arrow points in the direction of all the numbers that are greater than \(a\).
    The filled-in dot highlights that fact that \(x\) can be equal to \(a\):


  • \(x < a \) : \(x\) is less than \(a\).
    This is represented by an arrow with an empty dot above the \(a\).
    The arrow points in the direction of all the numbers that are less than \(a\).
    The empty dot highlights that fact that \(x\) cannot be equal to \(a\):


  • \(x \leq a \) : \(x\) is less than or equal to \(a\).
    This is represented by an arrow with an filled-in dot above the \(a\).
    The arrow points in the direction of all the numbers that are less than \(a\).
    The filled-in dot highlights that fact that \(x\) can be equal to \(a\):

Example

Illustrate the solutions to each of the following inequalities using the number line:

  1. \(x \leq 4 \)

  2. \(x > -3 \)

  3. \(x < 6 \)

  4. \(x \geq 1\)

  5. \(x \leq -1 \)

  6. \(x > 2 \)

Solution +

Solution

  1. The solutions to \(x \leq 4 \) are all numbers less than or equal to \(4\):


  2. The solutions to \(x > -3 \) are all numbers greater than \(-3\):


  3. The solutions to \(x < 6 \) are all numbers less than \(6\):


  4. The solutions to \(x \geq 1 \) are all numbers greater than or equal to \(1\):


  5. The solutions to \(x \leq -1 \) are all numbers less than or equal to \(-1\):


  6. The solutions to \(x > 2 \) are all numbers greater than \(2\):