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Inequality question

  • Thread starter Brady
  • Start date
  • #1
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A pretty simple problem, but I'm confused nonetheless. :yuck:

|x-8|<|2x+1|

Help would be greatly appreciated.
 

Answers and Replies

  • #2
18
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In addition, write |-5x^5y^2| two ways.
 
  • #3
Tide
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Homework Helper
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You can start by sketching a graph of each side and comparing them. That should give you some major insights!
 
  • #4
18
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I've pretty much spent hours working on this problem; for some reason, it's still not getting to me. False contradiction and confusion keep popping up...argh
 
  • #5
18
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please help me
 
  • #6
Tide
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Did you make a sketch?
 
  • #7
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..? Didn't I just explain that I tried this problem for a terriby long time, and I would like a clear explanation?
 
  • #8
37
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Don't we deserve to see your approach?

Brady said:
..? Didn't I just explain that I tried this problem for a terriby long time, and I would like a clear explanation?
Brady said:
A pretty simple problem, but I'm confused nonetheless. :yuck:

|x-8|<|2x+1|

Help would be greatly appreciated.
|x-8| - |2x+1| < 0

This solution is based on the fact that if a continuous function changes its sign, it might happen only in this function zeroes.

1.
Mark zeroes for each |expression| on the number line: x=8 and x=-1/2.
This two points break the whole number line into three intervals:
I. {-inf, -1/2}: -inf < x < 8
II. [-1/2, 8}: -1/2 <= x < 8
III. [8, +inf}: 8<= x < +inf.

Each expression "keeps" its sign unchanged on each interval.
Draw the number line and put the expressions' signs on the intervals
x-8: .............-..........|........-.........|.......+.........
2x+1: ...........-..........|........+.........|.......+.........

Solve inequality on each region separately following the definition
|a| = -a, if a<0
|a| = a, if a>=0

I.
x-8<0 and 2x+1<0 therefore
-(x-8) - (-(2x+1)) < 0
-x + 8 + 2x + 1 < 0
x < -9 - agrees with -inf < x < 8 condition.

II.
x-8<0 and 2x+1>=0 therefore
-(x-8) - (2x+1) < 0
-x + 8 - 2x - 1 < 0
3x > 7
x > 7/3 - considering -1/2 <= x < 8 restriction
the answer on this interval
7/3 < x < 8

III.
x-8>=0 and 2x+1>=0 therefore
x - 8 - (2x+1) < 0
x - 8 - 2x - 1 < 0
x > - 9 - considering x>=8 restriction
the answer on this interval
x >= 8.

Combininig all the answers,
x < -9 or x > 7/3.

Faster way - just like Tide recommended.
Draw two graphs:
f(x) = |x-8| and g(x) = |2x+1|
(do you need help with plotting them?).

|x-8|<|2x+1| is equvalent to
graph f(x) is below graph g(x).
These two graphs intersect at x=-9 and x=7/3.
Graph f(x) is below graph g(x) left of -9 and right of 7/3.
The same answer.
 
  • #9
HallsofIvy
Science Advisor
Homework Helper
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911
Tide said:
You can start by sketching a graph of each side and comparing them. That should give you some major insights!
Brady said:
please help me
Tide just did!

Tide said:
Did you make a sketch?
Brady said:
..? Didn't I just explain that I tried this problem for a terriby long time, and I would like a clear explanation?
Yes, you did. You did not, however, say whether or not one of the things you tried was graphing the two functions as Tide suggested- that was his question.
In fact, although you say you tried, you have shown us NOTHING of what you tried.
That is, after all, a requirement for homework help on this forum.
 

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