We have learned a lot of new material recently. We talked about linear equations. We talked about inequalities and absolute value. Now let's review.
One useful way to represent a line is using the point slope form. If we know the slope of a line, and a single point that the line goes through, we can use this formula to express the equation of a line. Let's say you have a line that goes through the point 4, 7 with slope of negative 3. What is the equation of that line?
Well, the point slope formula takes a point written as x1y1 and a slope m. The equation is then, y minus y1 equals m times the quantity x minus x1. In this situation our point is 4,7 and m is negative 3. So the answer is y minus 7 equals negative 3 times the quantity x minus 4.
We also learned how to simplify equations by pulling out a common factor. Can you factor out the common term in this expression?
The 8 and 12 share 4 as a common factor. They each share an x and a y as well. So the factored version of this would be, 4xy times the quantity 2x plus 3y.
We learned that absolute value really represents the difference between two numbers. How would you write the following sentence as an absolute value equation? The distance between x and 8 is less than 3.
The distance between x and 8 is represented as the absolute value of x minus 8. Since that must be less than 3, the solution is absolute value of x minus 8 is less than 3.
Looking at this statement, the absolute value of x minus 8 is less than 3. How would you write the solution to this inequality using interval notation?
We can write negative 3 as less than x minus 8 which is less than 3. Adding 8 gives us 5 is less than x, which is less than 11.
We also saw that some equations which don't initially look linear actually are. Can you solve the following equation for x?
Noting that x cannot equal negative 4 and x can also not equal 0, since that would make the denominator 0. Multiply both sides by the denominators, the quantity 4 plus x and 2x, not forgetting the parentheses. Distribute. Subtract divide by negative 27. And the answer is 4 over 27.