- #1

- 18

- 0

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

Help would be greatly appreciated.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Brady
- Start date

- #1

- 18

- 0

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

Help would be greatly appreciated.

- #2

- 18

- 0

In addition, write |-5x^5y^2| two ways.

- #3

Tide

Science Advisor

Homework Helper

- 3,089

- 0

- #4

- 18

- 0

- #5

- 18

- 0

please help me

- #6

Tide

Science Advisor

Homework Helper

- 3,089

- 0

Did you make a sketch?

- #7

- 18

- 0

- #8

- 37

- 0

Brady said:

Brady said:

|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

- 41,847

- 967

Tide said:

Tide just did!Brady said:please help me

Tide said:Did you make a sketch?

Brady said:

Yes, you did. You did

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.

Share: