Inequalities

Definition of Inequalities
An inequality is the relationship between two expressions; can be equal or non-equal, and greater than (more than) or less than (fewer than).

• ‘Solving’ an inequality means finding all of its solutions.
• A “solution” of an inequality is a number which when substituted for the variable makes the inequality a true statement.

Symbols with description of Inequalities

Graphing Symbols with description of Inequalities

               The closed circle indicates that this is Equal to the numeral graphed.

Solving

How to Solve?

Solving inequalities is very like solving equations. But we must also pay attention to the direction of the inequality.

Direction : Which way the arrow “points”

Some things we do will change the direction!

                             

Safe things to do

These are things we can do without affecting 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

Example 1

Example 2

Here are the examples with solutions :

Adding or Subtracting a Value

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

Example

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality.

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

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value. But we need to be a bit more careful.

Positive Values

Everything is fine if we want to multiply or divide by a positive number :

Example

Negative Values

When we multiply or divide by a negative number, we must reverse the inequality.

Why?

Look at the number line!

Example

Example

Note : I reversed the inequality on the same line and I divided by the negative number.

Remember : When multiplying or dividing by a negative number, reverse the inequality.

Multiplying or Dividing by Variables

Example

So do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

References :

• http://www.sosmath.com/algebra/inequalities/ineq01/ineq01.html

• http://www.mathsisfun.com/algebra/inequality-solving.html