Hi
We are going to start solving inequalities (inecuaciones)
First of all, some basic instructions from Mathisfun:
Solving Inequalities
Sometimes we need to solve Inequalities like these:
|
Symbol |
Words |
Example |
|---|---|---|
| > |
greater than |
x + 3 > 2 |
| < |
less than |
7x < 28 |
| ≥ |
greater than or equal to |
5 ≥ x - 1 |
| ≤ |
less than or equal to |
2y + 1 ≤ 7 |
Solving
Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:
| Something like: | x < 5 | |
| or: | y ≥ 11 |
We call that “solved”.
How to Solve
Solving inequalities is very like solving equations … we do most of the same things …
… but we must also pay attention to the direction of the inequality.
Direction: Which way the arrow “points”
Some things we do will change the direction!
< would become >
> would become <
≤ would become ≥
≥ would become ≤
Safe Things To Do
These are things we can do without affecting the direction of the inequality:
- Add (or subtract) a number from both sides
- Multiply (or divide) both sides by a positive number
- Simplify a side
Example: 3x < 7+3
We can simplify 7+3 without affecting the inequality:
3x < 10
But these things will change the direction of the inequality (”<” becomes “>” for example):
- Multiply (or divide) both sides by a negative number
- Swapping left and right hand sides
Example: 2y+7 < 12
When we swap the left and right hand sides, we must also change the direction of the inequality:
12 > 2y+7
What If I Solve It, But “x” Is On The Right?
No matter, just swap sides, but reverse the sign so it still “points at” the correct value!
Example: 12 < x + 5
If we subtract 5 from both sides, we get:
12 - 5 < x + 5 - 5
7 < x
That is a solution!
But it is normal to put “x” on the left hand side …
… so let us flip sides (and the inequality sign!):
x > 7
Do you see how the inequality sign still “points at” the smaller value (7) ?
And that is our solution: x > 7
Note: “x” can be on the right, but people usually like to see it on the left hand side.
Negative Values
 |
When we multiply or divide by a negative number we must reverse the inequality. |
Why?
Well, just look at the number line!
For example, from 3 to 7 is an increase,
but from -3 to -7 is a decrease.
 |
See how the inequality sign reverses (from < to >) ?
Let us try an example:
Solve: -2y < -8
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
(Note that I reversed the inequality on the same line I divided by the negative number.)
So, just remember:
When multiplying or dividing by a negative number, reverse the inequality
2. If these explanations weren’t clear enough, go to this link, pages 33, 34 and 35.
3. Go here, and try to do as many exercises as possible. You can read the examples and explanations, as well.
