Multiplying or Dividing by Variables
Here is another (tricky!) example:
Solve: bx < 3b
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 we 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).
A Bigger Example
Solve: (x-3)/2 < -5
First, let us clear out the "/2" by multiplying both sides by 2.
Because we are multiplying by a positive number, the inequalities will not change.
(x-3)/2 ×2 < -5 ×2
(x-3) < -10
Now add 3 to both sides:
x-3 + 3 < -10 + 3
x < -7
And that is our solution: x < -7
Two Inequalities At Once!
How do we solve something with two inequalities at once?
Solve:
-2 < (6-2x)/3 < 4
First, let us clear out the "/3" by multiplying each part by 3:
Because we 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 we 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
Summary
- Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.
- But these things will change direction of the inequality:
- Multiplying or dividing both sides by a negative number
- Swapping left and right hand sides
- Don't multiply or divide by a variable (unless you know it is always positive or always negative)
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