Linear Inequalities / Quadratic Inequalities

hello there!

first off we hand in our homework due today!Ex 27 #11

=The sum of two numbers is 181. Three times the larger plus twice the smaller equals 459. Find the numbers

1. we need to set up the equation
x + y = 181
3x+2y=459
2. eliminate by multiplying x + y = 181
2(x + y)=2(181)
2x+2y=362
3. subtract
3x+2y=459
-2x+2y=362
so you'll get x = 97
4. substitute the x=97 to the equation 2x+2y=362 and find y
2(97)+2y=362
194 + 2y=362
2y=362-194
2y=168
y=84

1. we were given an example

x+y>6
2x+2y<5>

"and" means intersection
=we have to look for x and y plane where both equations are true.

the red broken line represents the 1st equation.
the black broken line represents the 2nd equation.


if we were to check the intersection of the points we need to find a points that satisfy the 2 equation.
(0,0) is the easiest point to check if the line is not touching it!


so if we substitute that in the solution we will get:

0+0>6 that's true
2(0)+2(0)<5>the shaded part represents the intersection of the 2 equation.

*remember that ><>/\<>and included*









"or" means union
=we have to find solution that satisfy 1 of the equation

x-2y<5>3

(again this line will be excluded and will be broken line)

the green line represents the 1st equation.

the blue line represents the 2nd equation.

we need to find each quadrant were one of the points satisfy one equation.
we also need to check each quadrant. we use the points (1,-5) where only one of the equation was satisfied.

we are finished on this topic. Now were moving on to the new one, its all about Quadratic Inequalities.

Speaking of quadratic, what image do you think will be using??the answer is the parabola.

equation will be *y=axsquared+bx+c*

1st step is to graph:
this is a general parabola, we have to take a point inside and outside the parabola so we know which part should be shaded.
















we are given an example:

y>/x(squared)-2x-8

-we have to find the vertex
b/2a=-2/2=-1
x=1

-then substiture
y=1-2-8
y=-9

so the vertex will be (1,-9)

then we need to find the intercepts
x(squared)-2x-8
(x+4)(x-2)
intercepts are (4,-2)


then we need to find a point inside and
outside of the parabola. (0,0) for inside, and (5,0) for the outside.

check:
0>/0-8(inside)
5>/5-8(outside)

next example is x(squared)-2x-8>/0

we use solid circle because the points are included, we use this in set notations.

*the blue line represents the outside points. [-2,-infinity] and [4,infinity].

*the maroon line represents the inside points, points between -2 to 4.

In this case we only have one dimension (x-axis) so you can drop the y-axis. We don't have to worry about it.

Again we always have to check if were right, we have to find a point that satisfy the equation.

(0,0) can't be the right point because if we substitute it we'll get 0-0-8>/0. Its wrong!

(5,0) we'll get 5+10-8>/0. Its right!

(6,0) we'll get 36-12-8>/0. Its right too!

So either one of the two points can be use to satisfy the equation but not (0,0)

____________________________________________________________________

Sorry because I'm not sure how to do the uppercase of 2 so i put squared!
The shaded part represents the intersection.

Oh don't forget we have homework to do! Exercise 28 # 1-3

hope you did learn something on my blog!!..good luck to everyone!!:))

cheers!!!


the next scribe will be Kim C.:D..