How do I solve absolute value inequalities involving polynomials?

[moondaaay]

1.

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Homework Equations

Solving Polynomial Inequalities

The Attempt at a Solution

Then I used the property of absolute value inequality to get rid of it.

But I really don't know if I'm doing the right step. Is this correct? So that I could separate them in two cases and find the solution set.

 

[andrewkirk]

Divide it into two alternative cases.
(1) It's equal to zero. This is the easiest one. What can we say about x in that case?
(2) It's less than zero. What does that imply about the signs of the numerator and the denominator. Is there any part of either that we can ignore?

 

[SammyS]

1. View attachment 89960

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Homework Equations

Solving Polynomial Inequalities

The Attempt at a Solution

View attachment 89960
Then I used the property of absolute value inequality to get rid of it.

But I really don't know if I'm doing the right step. Is this correct? So that I could separate them in two cases and find the solution set.
View attachment 89962

Hello moondaaay. Welcome to PF !

Where is the function on the left side of the inequality zero?

What is its domain?

 

[symbolipoint]

The inequality has four critical values for x, splitting the real number line into five intervals. Each interval can be checked to find which are solutions to the inequality and which are not solutions.

 

[moondaaay]

The inequality has four critical values for x, splitting the real number line into five intervals. Each interval can be checked to find which are solutions to the inequality and which are not solutions.

But what should I do first to remove the absolute value?

 

[symbolipoint]

But what should I do first to remove the absolute value?

Not really important here. Whatever sign the 3x+7, its absolute value will still be POSITIVE OR ZERO.