Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Solving Problems Involving  Inequalities 10-3

For practice, we went through the examples in our textbook on Page 469
We discovered that reading and re-reading the problem was critical to make sure we answered the exact question.  As noted in Example 1: the question asked what is the minimum total distance, to the nearest mile, that she will have to travel...?"  The critical part, was "to the nearest mile."  Please read the problem and then realize  why re arrived at the following:
Let d = the distance fro the sign to home
d - 16 > 25
solving that open sentence we get
d > 41
Since the distance needs to be greater than 41, the next whole number is 42 so the answer is
The minimum distance she will travel is 42 miles.

To translate phrases such as "is at least" and "is no less than"  you will need   ≥ 
...think I want at least $200 when going to Disneyland. I obviously want more.. but I will be happy with $200.

To translate phrases such as "is at most" and "is no more than"  you will need   ≤
...think I want at most 7 problems of homework. I really want fewer than 7 but I'll be okay with 7.

Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Solving Inequalities 10-2


Property of Comparison
For all real numbers a and b, one and only one of the following statements is true:

a < b,            a = b,       or a >; b

Transitive Property of Order
For all real  numbers a, b, and c :

1. If a < b and b < c , then  a <; c
2. If a > b, and b > c, then a > c

Addition Property of Order
For all real  numbers a, b, and c :

1. If  a < b, then a + c < b + c
2. If a > b, then a + c > b + c

Multiplication Property of Order


For all real  numbers a, b, and c such that
c > 0 ( c is positive)

1. If a < b, then ac < bc
2. If a > b then ac > bc

c < 0 ( c is negative)

1. If a < b, then ac > bc
2. If a > b, then ac < bc

Multiplying each side of an inequality by a negative number reverses the direction or order of the inequality.

These properties guarantee that the following transformations of a given inequality will always produce an equivalent inequality

1. Substituting for either side of the inequality an expression equivalent to that side
2. Adding to (or subtracting from) each side of the inequality by the same real number
3. Multiplying (or dividing) each side of the inequality by the same positive number.
4. Multiplying  or dividing) each side of the inequality by the same negative number AND  reversing the direction of the inequality. 

To solve an inequality you use the same steps used to solve equations

1. Simplify each side of the inequality as needed
2. Use the inverse operations to undo any additions or subtractions
3. Use the inverse operations to undo any multiplications or divisions

Remember when graphing these inequalities a closed dot indicates that the number is included while an open dot indicates a less than or a greater than.

When working with inequalities it is important to read both the symbols and the words slowly and carefully!!

Word problems involving inequalities also require very careful reading. An important step to solving this type of problem is determining which inequality symbol to use. Before you write your inequality you should be certain that you will use the correct symbol.

Algebra Honors (Period 6 & 7)

Inequalities in One Variable
Order of Real Numbers 10-1

This sections should be review... you have been working with less than and greater than symbols since 6th grade. A number line shows order relationships among all real numbers. The value of a variable may be unknown but you may know that is either greater than or equal to another number.
For example,
x ≥ 5 is read "x is greater than or equal to 5."
x ≥ 5 is another way of writing " x >5 or x = 5"

Translating statements into symbols is a critical concept. Practice these as review
-3 is greater than -5.    -3 > -5
and
x is less than or equal to 8  x    ≤   8

To show that x is between -4 and 2 you write
-4 < x < 2
which is read 
"-4 is less than x and x is less than 2."
or you could read it as
" x is greater than -4 AND less than 2."

The same comparisons are stated in the sentence  2 > x  > -4

When all the numbers are know you can classify the statement as true or false.
Thus,
-4 < 1 < 2   is true
but
-4 < 8 < 2 is false

An inequality is formed by placing an inequality symbol (  > ,  < ,     ≤ , or    ≥) between numerical pr variable expressions-- called the SIDES of the inequality
You solve an inequality by finding the values from the domain of the variable which make the inequality a true statement.   Such values are called the solutions of the inequality. All the solutions make up the solution set of the inequality.