lesson park

Inequalities

Lesson park

Definition

• An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value

• a ≠ b says that a is not equal to b

• a < b says that a is less than b

• a > b says that a is greater than b

Example

• Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

  
  

• Something like: x < 5 or: y ≥ 11

  
  

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  We call that "solved".

  
  

  Example: x + 2 > 12Subtract 2 from both sides:

  x + 2 − 2 > 12 − 2

  
  

• Simplify:

  x > 10

  Solved!

Some things can change the direction!

• < becomes >

• > becomes <

• ≤ becomes ≥

• ≥ becomes ≤

Safe Things To Do

These things do not affect the direction of the inequality:

• Add (or subtract) a number from both sides

• Multiply (or divide) both sides by a positive number

• Simplify a side

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Example:

• 3x < 7+3

• We can simplify 7+3 without affecting the inequality:

• 3x < 10

Adding or Subtracting a Value

• We can often solve inequalities by adding (or subtracting) a number from both sides

• Example: x + 3 < 7

• If we subtract 3 from both sides, we get:

• x + 3 − 3 < 7 − 3   

  x < 4

  
  

• And that is our solution: x < 4

  In other words, x can be any value less than 4

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

• If we subtract 5 from both sides, we get:

• 12 − 5 < x + 5 − 5    

  7 < x

• That is a solution!

  But it is normal to put "x" on the left hand side ...

  ... so let us flip sides (and the inequality sign!):

• x > 7

• Do you see how the inequality sign still "points at" the smaller value (7) ?

  And that is our solution: x > 7

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