An inequality is similar to an equation. There are two expressions separated by a symbol that indicates how one expression is related to the other. In an equation such as 7x = 49, the = sign indicates that the expressions are equivalent. In an inequality, such as 7x > 49, the > sign indicates that the left side is larger than the right side.
Sometimes we need to solve Inequalities like these:
Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:
Something like:
or:
x < 5
y ≥ 11
We call that "solved".
Let us divide both sides by -2 ... and reverse the inequality!
-2y < -8
-2y/-2 > -8/-2
y > 4
And that is the correct solution: y > 4
It seems easy just to divide both sides by b, which would give us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one!
To help you understand, imagine replacing b with 1 or -1 in that example:
if b is 1, then the answer is simply x < 3
but if b is -1, then you would be solving -x < -3, and the answer would be x > 3
So:
Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).
To solve the inequality 7x > 49, we follow the same rules that we did for equations. In this case, divide both sides by 7 so that x > 7. This means that x is a value and it is always larger than 7, and never equal to or less than 7.
The "less than" symbol (<) may also be seen in inequalities.
Example:
500 + x < 1200
500 + x - 500 < 1200 - 500
0 + x < 700
x < 700
-2 < (6-2x)/3 < 4
First, let us clear out the "/3" by multiplying each part by 3:
Because you are multiplying by a positive number, the inequalities will not change.
-6 < 6-2x < 12
Now subtract 6 from each part:
-12 < -2x < 6
Now multiply each part by -(1/2).
Because you are multiplying by a negative number, the inequalities change direction.
6 > x > -3
And that is the solution!
But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):
-3 < x < 6