A linear inequality is a mathematic statement that
contains an inequality symbol. The
inequality symbols that you will see in this lesson are|
|
> |
greater than (the quantity to the left of the sign is
greater than the quantity to the
right of the sign) |
|
< |
less than (the quantity to the left of the sign is
less than the quantity to the right of
the sign) |
|
≥ |
greater than or equal to (the quantity to the left of
the sign is greater than or equal
to the quantity to the right of the sign) |
|
≤ |
less than or equal to (the quantity to the left of
the sign is less than or equal to the
quantity to the right of the sign) |
Your first task is to determine whether or not a number makes an inequality
true.
Example 1: Determine which elements of 
satisfy 6x + 7 > 8 .
Next, you’ll need to be able to solve inequalities and
then graphy your solution on a
number line graph .
To solve an inequality, you can add or subtract the same quantity to both sides
of the
inequality, or you can multiply or divide both sides of the inequality by the
same number.
However, if you multiply or divide both sides of an inequality by a negative
number , you
must reverse the direction of the inequality.
For each of the fol lowing examples , solve the inequality and then graph your
solution on
the real number line.
Example 2: 7x > 35
Example 3: - 4x ≤ 48
Example 4: 3x - 4 ≤11
Example 5: 5x - 4 > 2x + 7
Example 6: - 2(x - 5) ≤ 3(4x - 7) +12
Next, we’ll solve linear inequalities in two variables by
graphing .
The answer to these problems will be the half-plane on one side of a dotted or
solid line
that you will graph.
Example 7: Solve: x ≤ 3
Example 8: Solve: y < -2
Example 9: Solve y ≥ 2x -1
Example 10: Solve: 2x - 3y > 6
Next we’ll review of couple of topics from elementary
algebra ;
Multiplying Binomials
You should be able to multiply two binomials together. Many students use the
acronym
FOIL to help them accomplish this task.
Example 11: Multiply: (2x - 3)(x - 5)
Example 12: Multiply: (x + 6)(x -1)
Example 13: Multiply: (2x -1)2
Example 14: Multiply: (3x - 2)(3x + 2)
Next, you should recall how to factor. To factor, you’ll
want to break up the polynomial
you are given into two binomials that you would multiply together to get the
given
polynomial.
Example 15: Factor: x2 -8x - 20
Example 16: Factor: x2 - 5x + 6
Example 17: Factor: x2 + x - 6
Example 18: Factor: x2 -16
Example 19: Factor: x2 -10x + 25
Example 20: Factor: x2 + 3x -10
Example 21: Factor: x2 + 7x +12
Example 22: Factor completely : 3x2 +12x + 9
Sometimes you will need to factor by grouping .
Example 23: Factor: x2 + 3x + ax + 3a
Example 24: Factor: xy + 3x + 4y +12
Example 25: Factor: x3 - 3x2 - 3x - 27
You will need to be able to factor polynomials where the first number is not 1.
You can
use either trial and error, the box method that some students have learned, or
factoring by
grouping. I’ll use FBG in class
Example 26: Factor: 6x2 - x -12
Example 27: Factor: 10x2 -13x - 3
Example 28: Factor: 3x2 -10x -8
Example 29: Factor: 21x2 - 29x -10
Example 30: Factor: 12x2 - 4x - 5
