# Solve for x |-1/(2+x)|<=1/6

Remove the absolute value term. This creates a on the right side of the inequality because .
Set up the positive portion of the solution.
Solve the first inequality for .
Move to the left side of the equation by subtracting it from both sides.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Combine.
Reorder the factors of .
Combine the numerators over the common denominator.
Simplify the numerator.
Factor out of .
Reorder the expression.
Reorder and .
Reorder and .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Subtract from both sides of the equation.
Subtract from both sides of the equation.
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Consolidate the solutions.
Find the domain of .
Set the denominator in equal to to find where the expression is undefined.
Subtract from both sides of the equation.
The domain is all values of that make the expression defined.
Interval Notation:
Interval Notation:
Use each root to create test intervals.
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is less than the right side , which means that the given statement is always true.
True
True
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is greater than the right side , which means that the given statement is false.
False
False
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is less than the right side , which means that the given statement is always true.
True
True
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
The solution consists of all of the true intervals.
or
or
Set up the negative portion of the solution. When solving the negative portion of an inequality, flip the direction of the inequality sign.
Solve the second inequality for .
Move to the left side of the equation by adding it to both sides.
To write as a fraction with a common denominator, multiply by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Combine.
Reorder the factors of .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Multiply by .
Simplify with factoring out.
Rewrite as .
Factor out of .
Factor out of .
Move the negative in front of the fraction.
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Subtract from both sides of the equation.
Multiply each term in by
Multiply each term in by .
Multiply .
Multiply by .
Multiply by .
Multiply by .
Subtract from both sides of the equation.
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Consolidate the solutions.
Find the domain of .
Set the denominator in equal to to find where the expression is undefined.
Subtract from both sides of the equation.
The domain is all values of that make the expression defined.
Interval Notation:
Interval Notation:
Use each root to create test intervals.
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is greater than the right side , which means that the given statement is always true.
True
True
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is less than the right side , which means that the given statement is false.
False
False
Test a value on the interval to see if it makes the inequality true.
Choose a value on the interval and see if this value makes the original inequality true.
Replace with in the original inequality.
The left side is greater than the right side , which means that the given statement is always true.
True
True
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
The solution consists of all of the true intervals.
or
or
Set up the intersection.
and ( or )
Use the rule to find the intersection.
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Solve for x |-1/(2+x)|<=1/6

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