Question 1209364: What values of x satisfy |x - 4| + 2(x + 3) <= 11 + 5|x + 7| + 3x + 8.
Express your answer in interval notation.
Answer by greenestamps(13215)   (Show Source):
You can put this solution on YOUR website!
Simplify the inequality....
The behavior of the function changes when the arguments of the absolute value expressions are equal to 0 -- at x=4 and x=-7. That divides the number line into three intervals: (-infinity,-7], [-7,4], and [4,infinity)
Find the value on each interval that satisfy the inequality.
(1) (-infinity,-7]
On this interval,  and 
Of the values of x on the given interval (-infinity,-7], the ones that satisfy the inequality are those less than or equal to -26/3.
First part of solution set: (-infinity,-26/3]
(2) [-7,4]
On this interval, and 
Of the value of x on the given interval [-7,4], the ones that satisfy the inequality are those greater than or equal to -44/7.
Second part of solution set: [-44/7,4]
(3) [4,infinity)
On this interval,  and
All of the values of x on the given interval [4,infinity) are greater than or equal to -52/5.
Third part of solution set: [4,infinity)
ANSWER: the complete solution set is (-infinity,-26/3] U [-44/7,infinity)
The simplified form of the inequality I used is , which is equivalent to  .
Here is a graph of that function showing the value is less than or equal to 0 on (-infinity,-26/3] U [-44/7,infinity).
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