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.
However, the inequality sign reverses when:
- multiplying by a negative
- dividing by a negative
- taking the reciprocal of both sides
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:
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 .
In several cases, it is better to check the solutions to make sure that they satisfy the inequations. Not all solutions are feasible.
Think: why couldn't all solutions be feasible?