## Inequations Signs and Operations

means greater than

means greater than or equal to

means less than

means less than or equal to

In order to solve inequations, we need to see what effect one operation applied to both sides has on the inequality sign.

For example,

However, the **inequality sign reverses** when:

- multiplying by a negative
- dividing by a negative
- taking the reciprocal of both sides

For example:

## Solving Inequations

Inequations, just like equations are used to find unknown values of a pronumeral. For example,

However, what does mean?

This means that the solution is correct for all values of x **greater or equal to** 2.

If the solution is to be plotted on a number line, then the solution would like the following:

As seen above, the number 2 is circled fully (meaning that 2 is also part of a solution), as well as 3, 4, 5, 6, 7, and so on are calid solutions that satisfy the inequation.

*Remember:* fill the circle in if the solution is greater/less than or equal to () but do NOT fill the circle in if the solution is only greater/less than ().

Another kind of inequation as shown below:

The solutions are limited to a small region as shown in the number line below:

## Absolute Values

An absolute value is the magnitude of a real number without regard to its sign. The absolute value of any number is given by:

It could simply be understood as how far a number is from zero. For example, plot on a number line and evaluate *x*.

This means the distance of *x* from zero is 2 (in either direction).

However, what happens when or ?

means the distance from zero to *x* is less than or equal to 2 (in either direction). The other way to write this as a statement is .

On the other hand, means that the distance from zero to *x* is greater than 2 (in either direction). The other way to write this statement is .

To conclude,

### Examples:

Solve:

1.

### Show Answer

This means that the distance from zero to *x+4* is 7 in either direction.

2.

### Show Answer

This means that the distance from zero to *2y-1* is less than or equal to 5 in either direction.

3.

### Show Answer

This means that the distance from *5b-7* to zero is greater than 3 in both direction.

In several cases, it is better to check the solutions to make sure that they satisfy the inequations. Not all solutions are feasible.

4.

### Show Answer

Let's check whether both solutions satisfy the inequation.

(i) Substitute *x=3* to thein equation, and check whether the LHS equals the RHS.

Since both sides have equal value, that means *x=3* **IS** a solution.

(ii) Substitute *x=1/5* and check.

The opposite sides have unequal values and therefore *x=1/5* **IS NOT** a solution.

*Think: why couldn't all solutions be feasible?*