Inequalities are the relation between two expressions that may not be equal. So often we deal with
for example, where we plug in any x and get out a corresponding y. Sometimes though we might want to know where
Here we want to know specific values (points) that make that statement true.
The point (2,5) -> 5>7
This is not true, so (2,5) is not a solution.
The point (5,11) -> 11>10
This is true, so (5,10) is a solution
Inequalities are very similar to equations! However, instead of the variable being equal to a solution, it is either greater than, less than, greater than or equal to, or less than or equal to the solution.
in the equation 2x-8=14,
we add 8 to both sides, reaching 2x=22,
therefore, we must now divide both side by 2, with x finally reaching 11.
In the inequality 2x-8<14,
we solve it the same exact way, as long as we keep the sign the same. We progress by adding 8 on both sides, which leads us to 2x<22, then we divide both sides by 2, leading us to x<11. This reads x is less than 11.
The only exception to keeping the signs the same is when we must solve an inequality in which we must divide by a negative number. In this case, we flip the sign to its opposite.
Here, we begin by adding 6 to both sides, reaching -4x>24. We then proceed by dividing -4 on both sides and flipping the sign when finished. We should now have x<6.
In order to understand why we flip the sign, in the comparison of two numbers, one is always greater. However, when we make both negative, the smaller number then becomes the larger number.
For example, between 4 and 5, 5 is the greater number.
On the other hand, between -4 and -5, -4 is the greater number.