The appalachian trail is roughly 2,180 miles long. about how long is the trail in kilometer?
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.
the first thing to do is to subtract 16 from both side of the equation:
16 - 16 - b = 7 - 16
the 16s on the left cancel out resulting in:
-b = -9
think of the -b as -1(9). this is multiplication. by doing the inverse operation (division) we can get rid of the negative on the b. we will divide both sides by -1.
this leaves a positive b. since both parts of the right side of the equation are negative it becomes positive ( a negative divided by a negative is a positive).
the final answer is b= 9.
check our work by plugging in the b value (9) into the original equation:
16 - 9 = 7
7 = 7 is true so we now we have the right answer.
24/4 = 6
20/4 = 5