## Week15- solving rational equations

Last week we learned how to simplify a rational expression, and this week we explored how to solve a rational equation.

The difference between equation and expression is pretty straightforward, the expression does not have an equal sign, it only expresses its meaning. An equation on the other hand obviously have an equal sign, and our intent will be the solution the unknown instead of just simply simplifying it.

There are two common ways to solve a rational equation.

1: Cross multiplication, only works when there is only one fraction on each side of the equal sign.

EX:

In this case, I multiplied(q-2)by5, and (q+4)by3.

2. Use a common denominator, and solve.

EX:

The common denominator, in this case, is 6z.

## Week14- Rational Expressions

Last week we learned about Rational Expression, aka all real values of the variable except for those values that make the denominator zero. AKA2: In x/y, how to make sure y is not 0.

In particular, we learned about the how to make sure the denominator not equal 0 with something called non-permissible(the numbers one can’t use, or the denominator will equal zero and the expression will no longer be Rational.)

EX:

Outside of non-permissible values, we also learned how to simplify and/or multiple, divide Rational Expressions. But due to the fact, a Rational Expression or any real number, in this case, is a fraction, all to be done is to follow the law of dividing and multiplying of the fractions. AKA: why diving, multiply by it’s reciprocal. and why multiplying just multiply the numerator with the numerator and the denominator and denominator, but just keep in mind that in any rational expression the denominator can’t be 0, so there may be non-permissible as well. And as it’s an expression, all you can do is to simplify it.

In a fraction, when denominator equals the numerator, the number equals 1. EX: 8/8=1

EX:

So in an equation such as 3x.8(x-3)/2(x-3).9x^2. It might be helpful to use factoring to find the common thing that can cancel out to 1, like how (x-3) and 3x in this case. So after finding the common terms. they will just cancel out to 1, and in this case, 3x and 9x^2,(x-3)and (x-3) cancels out. and leaves you with 8/2.3x=8/6x=4/3x and don’t forget about the non-permissible, and remember you have to find them in the original equation, as they will be lost when you simplify the equation. In this case(x-3)can’t be 0, therefore x not equal 3, and 9x^2 can’t be 0, therefore x can’t equal 0.

## Week 13- graphing linear reciprocal functions

a reciprocal of a number is the opposite of the number, or using the cool math language is 1/the number.

If you known fraction then you should know if 1/x and x<1, the number will become greater, and vice versa if x>1, the number will become smaller.

So in graphing, this will result in some very interesting observations.

Let’s use this linear function as an example, x+6=y.

when you invert this function to 1/x+6=y, magical things will happen, ex:1/6+6=1/12, whereas the original function will yield 12. But this sort of magic has three exceptions,  1,-1 and o, as their inverse equals themselves, expect 0, as 0 can’t be divided by. With this information in mind, we will be able to determine how a linear inverse function.

Because when 1and -1 inverse their yield doesn’t change, so when y=1,-1 those x coordinates are called invariant points.

The second step in solving is to draw something called vertical asymptote, as we are in pre-cal 11 the horizontal asymptote will be 0, the asymptote is like a great wall that keeps the number on their own space because 0 can’t be divided. the vertical asymptote will always be the roots or the x-intercept.

With the asymptote and invariant points, we can basically draw the function just follow close to the asymptote in the zone where the original function has been and cross over the invariant points, you have successfully drawn yourself an inverse linear function.

## Week 12- graphing linear absolute value function

An Absolute value (||) means whatever inside is going to be positive, when we put a linear function within ||, it becomes an absolute value linear function.

The graph of it will change, as we know, a linear function is a straight line, either having a 0 slope or / positive slope or \ slope.

When graphing an absolute value linear function, if the slope is not 0, the line will bounce back up in the opposite direction of the original function at x-intercept.

Why is that?

Let me show you an example:

2x+4=y, a simple straight line with a slope of 2, and y-intercept of 4.

But magic happens when you put||on to the function. As the output of an absolute value cannot be negative,  y, in this case, will also have to positive.  But with our previous graph of the original equation, the Y clearly dipped under 0.

Let’s see what happens if we put an x value that would make y<0 into the absolute value equation.  |2x(-4)+4|=4,

the Line reflected up, from this example we can learn that when graphing a linear absolute value function, all you need to do is find the x-intercept of the original line and draw a reflected line from the original root.

## Week 11- graphing linear inequality with two variables

In previous grades, we learned about linear equations and inequalities with one and two variables, now we learn how to solve them graphically.

To graph a linear inequality with two variables we firstly rearrange it to y-intercept form in order to graph it.

EX: 2y+6<4x,

With y-intercept form of the equation, we are able to graph a line just how we did in grade 9.

The difference is in an equality the equation represent the values on the line and inequality represent a whole area.

A line will divide the graph into two zones, we just need to determine which zone to shade(it represent a range of value)

The most accurate way of determining such things is by trial and error,

we would take two test coordinates from each section of the graph and plug it into the equation, if the equation is true then the side with the coordinates will be shaded.

EX: (0,0) , (-5,5)

2(0)+6<4(0) 6 is not less than 4, therefore the otherside will be shaded.

One last thing you have to do before you submit your answer is to determine whether you should use a solid line or dotted line if the sign involves = sign, you use a solid line.(In this case it’a dotted line due to the fact that it’s only a < sign).