Linear inequalities are incredibly similar to linear equations in the fact that they relate one variable to another variable.
Unlike equality, there are actually two kinds of inequality:
A strict inequality says that something is definitely less than or greater than something else. This is represented by < or >. This is like how a strict parent will make an absolute statement that something is definitely the case!
For instance, when we say that 3 < 4, we are actually saying that 3 is strictly less than 4.
We can also use this with variables to state something like x>5, which means that x is strictly greater than 5.
A weak inequality says that something is less than or equal to or greater than or equal to something else. Weak inequality is represented by ≤ or ≥. This is like how a parent who isn’t strict says that you could do something or you could do something else.
For instance, I could write x ≤ 9, which tells me that x must be less than or equal to 9, but that is a weaker statement than the strict statement that x<9, which tells me that x would have to be definitely less than 9.
An easy way to remember all of this notation is that the inequality symbol looks like it wants to eat the number that is larger, just like PacMan!
Of course, these kinds of statements are a bit boring all by themselves, so it is helpful to relate them to what we know about linear equations.
Linear inequalities usually take one of two forms:
The standard form of an inequality is Ax + By ≤ C, but we can replace the less than or equal to sign with any inequality we want! Just like for our linear equations, this form is not particularly useful, but does define a linear inequality.
The most common form of a linear inequality is the slope-intercept form. This looks like y ≥ mx + b, where m is the slope of the line, b is the y-intercept and x and y are both variables.
Think about what this inequality is actually saying and suddenly it becomes less cryptic! It states that y must be greater than OR equal to mx + b. This basically means that our y-value will always be on or above the line, we’ll show this idea in greater detail in the lesson on graphing linear inequalities. For now, we’ll have to be satisfied with the concept alone!
One other key difference between equality and inequality is how to operate using algebra on an inequality.
For equations, as long as you do the same operation to both sides of the equation, you’ll usually be fine and accurate in your math.
However, if you have an inequality, there is one operation that you have to be very careful about! Multiplying or dividing by a negative number results in the inequality flipping direction.
This means that if you have an inequality that looks like -3x ≥ 9, you should divide both sides by -3, but this creates the inequality x ≤ -3.
Most students know this rule and hate it, since it’s so easy to forget. However, have you ever thought about why on earth we have this rule in the first place?
What would happen if for -3x ≥ 9 we added 3x to both sides of the inequality?
We would get 0 ≥ 9+3x.
Subtract 9 from each side to find -9 ≥ 3x.
Divide both sides by 3 to find -3 ≥ x.
Wait a minute… This is the exact same answer we had before and we never flipped the sign! We did way more work, but hopefully you see that dividing (or multiplying) by a negative number is the same as swapping the sides of the inequality.
This is why we just change the direction of the inequality! Not to confuse students, but to save them all of that effort every time.
Try to build up your intuition for why we do things in mathematics. When a teacher tells you a rule, don’t follow it blindly; try to understand what you’re actually doing and you won’t forget these basic operations that allow you to solve harder problems.