S.O.S. Mathematics CyberBoard

Hello Everyone , i would really appreciate any of you to remind me of how to solve some problems , i really need to understand how it goes on as ive already have started universty .

First one is : Find the equation of the circle with centre (-2,3) that passes through (1,-1).

Second is : Find which values of x satisfy the following , 1/2-x<2

3. Express the following as either open or closed intervals
{x: |x-4|≤ 6}

4. Express the following intervals in the form { x: | ax + b | < c } or { x: | ax + b | ≤ c }

(-4,-2)


Thank you

First one: Find the equation of the circle with centre (-2,3) that passes through (1,-1).
The distance from the center, (-2,3), of the circle to the point (1,-1), which is on the circle, is equal to the radius of the circle. This gives:
.
Similarly, the distance from the center to any point, , on the circle is equal to the radius. Thus,
. Square both sides to get the usual form for the equation of a circle.



Of course, you will have a numerical value for .


Second one: Find which values of x satisfy the following, .

Method 1: Subtracting 1/2 from both sides gives: . Think about this: If the opposite (i.e. negative) of x is less than 3/2, then x itself must be greater than the opposite of 3/2.

Method 2: Adding x to both sides and subtracting 2 from both sides gives: . You can leave it like this or reverse the whole statement, including the sense (direction) of the inequality.

3. Express the following as either open or closed intervals.
The quantity, gives the distance that an arbitrary point x is from 4 on the number line. Your solution should be all the points which are a distance of 6 or less from 4. That should give you the answer.

To do this symbolically, think about what you would look for if you substitute various values in for x to get an answer by trial & error. You would need the absolute value of x-4 to be less than or equal to 6.
That means the absolute value of x-4 needs to be between -6 & 6 or equal to -6 or 6. Writing this symbolically, we have:
.
Add 4 to the left right AND middle of this inequality to get your result. Now convert your answer to interval notation.

4. Express the following interval in the form { x: | ax + b | < c } or { x: | ax + b | ≤ c } : (-4,-2)

Note: The parentheses, (), mean that and .

The interval is the set of points on the number line between -4 and -2 . This interval can be described as the set of points that are a distance of less than 1 from the point -3. Any point, x, on this interval must fulfill the condition: .

Hopefully, you can take it from there.