To add and subtract rational expressions, it is easiest to start with a common denominator. This can be found by factoring.

With this example

There needs to be a common denominator, so we can find what they both need or have in common. Since this is a binomial denominator, we can leave them in the factored forms.

We can then put all the numerator terms together to create one big fraction

Once the expression is simplified, we can go back to find the non-permissible values.

To do this, we can look at the original expression and find the number that, if replaced x, would result in a 0. If x = 2 and 2 is subtracted, that is 0, which is not allowed to be the numerator. If x is -2 and 2 is added, it becomes a 0. The non-permissible values for x are -2, 2

 

 

 

When multiplying and diving rational expressions, we can take the rational expressions and factor them, then find the restrictions, remove any common factors, and multiply through.

It is important to factor first then find the restrictions. Restrictions are there to find terms for the variables that could lead to having a 0 in the denominator. That is why restrictions are found by = the factor to 0 and finding the answer for the variable.

Next thing to note is that everything is always fully factored. This helps cancel out factors. After this, just multiply through. Factors can be left in factored form.

This week, we learned how to graph absolute values. Since absolute values mean the positive of the number, all numbers under 0 were reflected upwards.

Originally, this graph crossed -2 and went further down, since it is the absolute value, all the negative y values reflect to the positive.

So, we can say y = -2x -4 if x≤-2 and y = -(-2x -4) (the other possibility for an absolute value) if x>-2

the original line has an equal to because it includes the critical point of the middle point, whereas the reflected line does not include that number.

There is a certain part of the parabola that is reflected upwards. We can say that y = (x-4)² -5 if x≤1.8 and x≥6.2 and y=-((x-4)² -5) is 1.8<x<6.2

 

This week, we learned how to take a linear equation and quadratic equation and graph them to find a solution and taking 2 quadratics to find the solution.

When presented with two equations, y=-x+2 and y = (x – 4)² – 4, we can graph them to see where they intersect. The intersections will be their solutions.

Their solutions are (2,0) and (5,-3)

If we are given the equation y = (x – 4)² – 4 and y = (x – 6)² – 4, we can graph them to find their solutions.

They have 1 solution that is (5,-3)

Two equations can have 0 solutions if they never touch, 1 solution if they only meet in one spot, 2 solutions if they meet in two spots. They can have infinite solutions if they are the same equation.