Solving Equations and Inequalities
An equation consists of two mathematical expressions separated by an equals sign. For instance: 1+1=2. Any operations can be performed the same way on both sides of the equation and yield the same result. Inequalities consist of two mathematical expressions separated by a sign indicated which side is greater or lesser. For example: 1<2.
Equations and Inequalities
I’ve talked a little bit about equations so far, but now I want to give some definition to the term. An equation consists of two mathematical expressions separated by an equals sign. For instance, we might have an equation that says 1 plus 1 equals 2. This is true, because this value is the same as that value.
Now, the beauty of equations is that you can perform any operations the same way on both sides of the equation and the equation remains true. For instance, we can add 5 to both sides of this equation and the equation is still true. We can multiply both sides of the equation by 7 and the equation is still true.
We can divide both sides of the equation by 13 and this equation is still true. Anything you perform on both sides of the equation equally does not invalidate the equation. Similar to equations, we have what’s called an inequality. We might have the inequality 1 is less than 2.
This consists of a mathematical expression on one side and the mathematical expression on the other side and a sign that indicates which side is greater or lesser. That’s what we have here. This is read as 1 is less than 2. Similar to equations, we can perform the same operations on both sides of the inequality sign and it remains true.
We can add 5 to both sides and it’s still true. 6 is still less than 7. We can multiply both sides by 7 and it’s still true. This side is now 42 and this one is equal to 49. It’s still true. We could divide it by 13 and it’s still true. The one thing that we can’t do with inequalities without changing them that we can with equations is multiplying or dividing by a negative number.
If we wanted to multiply both sides of this inequality by -1, that would make it untrue, because in the most simple of examples if we have an equation that says 1 is less than 2, and you multiply both sides by -1, that becomes -1 is less than -,2 which is not true.
What you have to do if you multiply or divide both sides of an inequality by a negative number, what you have to do is reverse the inequality sign. This is now a valid inequality once again, because we switched this sign after multiplying by -1. To show in real simple terms what that looks like, if you have 1 is less than 2 and then you want to make it negative, you can say -1 now is greater than -2.
That is the true expression of the inequality. You can have greater than or less than, and you can also have a greater than or equal to, which is something of a combination of equations and inequalities.
We would write greater than or equal to as a greater than sign with a line under it. Similarly, less than or equal to would be less than with a line under it. This is kind of a more formal definition of equations and inequalities and what you can do with them.