now, we can always get check a sequence if it's geometric or arithmetic, by doing a quick division of two adjacent terms, or a quick difference.
the division of two adjacent terms will give use the common ratio if it's geometric.
the difference of two adjacent terms will give use the common difference if it's geometric.
so, low and behold, is an arithmetic sequence, whose common difference is -5/8, meaning to get the next term, we simply subtract that much from the current one.
answer: the solution is -5 < a < 1. the graph is attached below as an image.
let x = |a+2|. so that means we can replace the entire "|a+2|" part with just "x"
we go from this: 4 + |a+2| < 7
to this: 4 + x < 7
that's the same as x+4 < 7
solve for x by subtracting 4 from each side
x+4 < 7
x+4-4 < 7 - 4
x < 3
now replace x with |a+2| to get |a+2| < 3
from here, we use the rule that |x| < k breaks down into -k < x < k for some positive number k. in this case, k = 3 and x = a+2
|a+2| < 3
-3 < a+2 < 3 use the rule mentioned above
-3-2 < a+2-2 < 3-2 subtract 2 from all sides
-5 < a < 1
the solution for 'a' is the compound inequality -5 < a < 1 meaning that 'a' can be anything between -5 and 1. the value of 'a' cannot equal -5, nor can it equal 1.
to graph this, we draw out a number line. then plot two open circles at -5 and 1. shade between these open circles. do not fill in the open circles. they are not filled in to tell the reader "do not include the value as part of the solution". think of them as potholes in the road where you cannot drive.
check out the attached image below for the graph.
first, find the intersections of the parabolas using any method. i choose to use the elimination method
next, find the distance between the intersected points (3, -1) & (-1, 2):
d. 12, 6, 7
for the triangle to have non-zero area, the sum of the shortest two sides must exceed the longest side.
a: 3 + 7 = 10 . . triangle has zero area
b: 4 + 5 < 10 . . sides don't meet
c: 6 + 8 = 14 . . triangle has zero area
d: 6 + 7 > 12 . . these form a triangle with non-zero area.