Solving Simple Linear Inequalities with Integer Coefficients

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SOLVING SIMPLE LINEAR INEQUALITIES WITH INTEGER COEFFICIENTS

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The concepts for this exercise are summarized below. For a complete discussion, read the text.
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The only difference between a linear equation in one variable
and a linear inequality in one variable is the verb:
instead of an " = " sign, there is an inequality symbol (<, >, ≤, or ≥).

One new idea is needed to solve linear inequalities:
if you multiply or divide by a negative number,
then the direction of the inequality symbol must be changed.
This idea is explored in the current section. We begin with a definition:

DEFINITION: linear inequality in one variable
A linear inequality in one variable is a sentence of the form

ax+b< 0, a≠0 .
The inequality symbol can be any of the following: <, >, ≤, or ≥ .

Here is a precise statement of the tools for solving linear inequalities.
Try translating them yourself, before reading the discussion that follows.
Think: "What do these facts tell me that I can do?"

THEOREM: tools for solving inequalities
For all real numbers a , b , and c ,

a0 , then
abc .
The inequality symbol may be <, >, ≤, or ≥,
with appropriate changes made to the equivalence statements.

TRANSLATING THE THEOREM
The first sentence, " a<b     ⇔    a+c<b+ c " ,
says that you can add (or subtract) the same number to (or from) both sides of an inequality,
and it won't change the truth of the inequality.

Here's the idea:
if a lies to the left of b on a number line,
and both numbers are translated by the same amount c ,
then a+c still lies to the left of b+c .

The second sentence, " a0 .
This says that you can multiply (or divide)
both sides of an inequality by the same positive number,
and it won't change the truth of the inequality.

Here's the idea. Think about the situation when c=2 .
If a lies to the left of b on a number line,
and we double both number's distance from 0 ,
then 2a still lies to the left of 2b .

It's the third sentence where something really interesting is happening.
The third sentence, " abc " , holds for c<0 .
Notice the " < " symbol in the sentence " a " symbol in the sentence " ac>bc⁢ ".
This says that if you multiply (or divide)
both sides of an inequality by the same negative number,
then the direction of the inequality symbol must be changed
in order to preserve the truth of the inequality.

Let's look at an example to understand this situation.
In the sketch below, " a<b " is true,
because a lies to the left of b .
Multiplying both sides by -1 sends a to its opposite (-a)
and sends b to its opposite (-b).
Now, the opposite of a is to the right of the opposite of b .
That is, -a>-b .

This simple idea is the reason why you must flip the inequality symbol
when multiplying or dividing by a negative number.
SO REMEMBER!
When you multiply or divide both sides of an inequality by a negative number,
then you must change the direction of the inequality symbol!

EXAMPLES:
Solve: -6-3x ≥4
Solution:
Write a nice, clean list of equivalent sentences:

-6-3x ≥4	(original sentence)
-3x≥10	(add 6 to both sides)
x≤- 103	(divide both sides by -3 ; change the direction of the inequality symbol)

Solve: 2x < 5
Solution: x < 5/2

Solve: -3x > 7
Solution: x < -7/3

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