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Calculus/Absolute Value Inequalities

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Expert: - 12/5/2009


Question

Absolute Inequalities
My prolems are

|(x/3)-2|<=4

|2+(1/x)|>1

|x/(x+2)|<=2

I have solved linear, rational & Absolutes
but in these problems, anything over x is giving me tough time.
I have no idea on problem 3
Do you have any tips for me on these three forms.
Thank you.

Answer
Questioner: Mark
Country: United States
Category: Calculus
Private: No
Subject: Absolute Value Inequalities
Question:

My problems are

|(x/3)-2|<=4

|2+(1/x)|>1

|x/(x+2)|<=2

I have solved linear, rational & Absolutes
but in these problems, anything over x is giving me tough time.
I have no idea on problem 3
Do you have any tips for me on these three forms.
Thank you.  
....................................

When you handle no 2:

|2+(1/x)| > 1

You start with

2+(1/x) > 1   OR  2+(1/x) < -1

Handle each separately:


2 + 1/x  > 1

1/x  > - 1

Now if  x > 0 (pos), multiply through:

1 > - x

x > - 1, which means  x > 0, since that was our assumption.

But if  x < 0 (neg), multiply through:

1 < - x   << must switch.

x < - 1

So the union of these is  (- inf to -1 and 0 to +inf ), meaning all but
the interval from -1 to 0.
.........................
Now the other half:

2 + 1/x < -1

1/x < - 3

Now if  x > 0 (pos), multiply through:

1 < - 3x or  - 3x > 1

Divide:  (SWITCH)

x < -1/3

That is inconsistent.  So we get no positive solutions here.

But if  x < 0 (neg), multiply through:


1 > - 3x     << switched

3x > - 1

x > - 1/3

So we get ONLY the interval from -1/3 to 0.  [Remember, this part assumes x is negative.]

So here is the whole thing:

You get (-inf to -1) plus (-1/3 to 0) plus (0 to inf)

-------------------------------------------

|x/(x+2)|<=2


Now you will 'interpret':

| x/(x+2) | <= 2  to mean:

-2 <= x/(x+2) <= 2

You want to cross-multiply by (x+2), but take into account whether it is positive (x > -2) or negative (x < -2).

If the bottom is neg, cross-multiply, switching signs:

-2x - 4 >=  x >= 2x + 4

Write the two separate inequalities and solve.

......................

If the bottom is pos, cross-multiply, NOT switching:

-2x - 4 <=  x <= 2x + 4

Write the two separate inequalities and solve.

You will still have some thinking to do.  Good luck.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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