Finite Math Examples

Solve for x |(x-1)/(2x+3)|<1
Step 1
Write as a piecewise.
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Step 1.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 1.2
Solve the inequality.
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Step 1.2.1
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 1.2.2
Add to both sides of the equation.
Step 1.2.3
Subtract from both sides of the equation.
Step 1.2.4
Divide each term in by and simplify.
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Step 1.2.4.1
Divide each term in by .
Step 1.2.4.2
Simplify the left side.
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Step 1.2.4.2.1
Cancel the common factor of .
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Step 1.2.4.2.1.1
Cancel the common factor.
Step 1.2.4.2.1.2
Divide by .
Step 1.2.4.3
Simplify the right side.
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Step 1.2.4.3.1
Move the negative in front of the fraction.
Step 1.2.5
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 1.2.6
Consolidate the solutions.
Step 1.2.7
Find the domain of .
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Step 1.2.7.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.2.7.2
Solve for .
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Step 1.2.7.2.1
Subtract from both sides of the equation.
Step 1.2.7.2.2
Divide each term in by and simplify.
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Step 1.2.7.2.2.1
Divide each term in by .
Step 1.2.7.2.2.2
Simplify the left side.
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Step 1.2.7.2.2.2.1
Cancel the common factor of .
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Step 1.2.7.2.2.2.1.1
Cancel the common factor.
Step 1.2.7.2.2.2.1.2
Divide by .
Step 1.2.7.2.2.3
Simplify the right side.
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Step 1.2.7.2.2.3.1
Move the negative in front of the fraction.
Step 1.2.7.3
The domain is all values of that make the expression defined.
Step 1.2.8
Use each root to create test intervals.
Step 1.2.9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 1.2.9.1
Test a value on the interval to see if it makes the inequality true.
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Step 1.2.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.2.9.1.2
Replace with in the original inequality.
Step 1.2.9.1.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 1.2.9.2
Test a value on the interval to see if it makes the inequality true.
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Step 1.2.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.2.9.2.2
Replace with in the original inequality.
Step 1.2.9.2.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 1.2.9.3
Test a value on the interval to see if it makes the inequality true.
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Step 1.2.9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.2.9.3.2
Replace with in the original inequality.
Step 1.2.9.3.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 1.2.9.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 1.2.10
The solution consists of all of the true intervals.
or
or
Step 1.3
In the piece where is non-negative, remove the absolute value.
Step 1.4
Find the domain of and find the intersection with .
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Step 1.4.1
Find the domain of .
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Step 1.4.1.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.4.1.2
Solve for .
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Step 1.4.1.2.1
Subtract from both sides of the equation.
Step 1.4.1.2.2
Divide each term in by and simplify.
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Step 1.4.1.2.2.1
Divide each term in by .
Step 1.4.1.2.2.2
Simplify the left side.
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Step 1.4.1.2.2.2.1
Cancel the common factor of .
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Step 1.4.1.2.2.2.1.1
Cancel the common factor.
Step 1.4.1.2.2.2.1.2
Divide by .
Step 1.4.1.2.2.3
Simplify the right side.
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Step 1.4.1.2.2.3.1
Move the negative in front of the fraction.
Step 1.4.1.3
The domain is all values of that make the expression defined.
Step 1.4.2
Find the intersection of and .
Step 1.5
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 1.6
Solve the inequality.
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Step 1.6.1
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 1.6.2
Add to both sides of the equation.
Step 1.6.3
Subtract from both sides of the equation.
Step 1.6.4
Divide each term in by and simplify.
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Step 1.6.4.1
Divide each term in by .
Step 1.6.4.2
Simplify the left side.
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Step 1.6.4.2.1
Cancel the common factor of .
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Step 1.6.4.2.1.1
Cancel the common factor.
Step 1.6.4.2.1.2
Divide by .
Step 1.6.4.3
Simplify the right side.
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Step 1.6.4.3.1
Move the negative in front of the fraction.
Step 1.6.5
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 1.6.6
Consolidate the solutions.
Step 1.6.7
Find the domain of .
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Step 1.6.7.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.6.7.2
Solve for .
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Step 1.6.7.2.1
Subtract from both sides of the equation.
Step 1.6.7.2.2
Divide each term in by and simplify.
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Step 1.6.7.2.2.1
Divide each term in by .
Step 1.6.7.2.2.2
Simplify the left side.
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Step 1.6.7.2.2.2.1
Cancel the common factor of .
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Step 1.6.7.2.2.2.1.1
Cancel the common factor.
Step 1.6.7.2.2.2.1.2
Divide by .
Step 1.6.7.2.2.3
Simplify the right side.
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Step 1.6.7.2.2.3.1
Move the negative in front of the fraction.
Step 1.6.7.3
The domain is all values of that make the expression defined.
Step 1.6.8
Use each root to create test intervals.
Step 1.6.9
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 1.6.9.1
Test a value on the interval to see if it makes the inequality true.
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Step 1.6.9.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.6.9.1.2
Replace with in the original inequality.
Step 1.6.9.1.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 1.6.9.2
Test a value on the interval to see if it makes the inequality true.
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Step 1.6.9.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.6.9.2.2
Replace with in the original inequality.
Step 1.6.9.2.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 1.6.9.3
Test a value on the interval to see if it makes the inequality true.
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Step 1.6.9.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 1.6.9.3.2
Replace with in the original inequality.
Step 1.6.9.3.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 1.6.9.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 1.6.10
The solution consists of all of the true intervals.
Step 1.7
In the piece where is negative, remove the absolute value and multiply by .
Step 1.8
Find the domain of and find the intersection with .
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Step 1.8.1
Find the domain of .
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Step 1.8.1.1
Set the denominator in equal to to find where the expression is undefined.
Step 1.8.1.2
Solve for .
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Step 1.8.1.2.1
Subtract from both sides of the equation.
Step 1.8.1.2.2
Divide each term in by and simplify.
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Step 1.8.1.2.2.1
Divide each term in by .
Step 1.8.1.2.2.2
Simplify the left side.
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Step 1.8.1.2.2.2.1
Cancel the common factor of .
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Step 1.8.1.2.2.2.1.1
Cancel the common factor.
Step 1.8.1.2.2.2.1.2
Divide by .
Step 1.8.1.2.2.3
Simplify the right side.
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Step 1.8.1.2.2.3.1
Move the negative in front of the fraction.
Step 1.8.1.3
The domain is all values of that make the expression defined.
Step 1.8.2
Find the intersection of and .
Step 1.9
Write as a piecewise.
Step 2
Solve when .
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Step 2.1
Solve for .
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Step 2.1.1
Subtract from both sides of the inequality.
Step 2.1.2
Simplify .
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Step 2.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 2.1.2.2
Combine and .
Step 2.1.2.3
Combine the numerators over the common denominator.
Step 2.1.2.4
Simplify the numerator.
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Step 2.1.2.4.1
Apply the distributive property.
Step 2.1.2.4.2
Multiply by .
Step 2.1.2.4.3
Multiply by .
Step 2.1.2.4.4
Subtract from .
Step 2.1.2.4.5
Subtract from .
Step 2.1.2.5
Factor out of .
Step 2.1.2.6
Rewrite as .
Step 2.1.2.7
Factor out of .
Step 2.1.2.8
Rewrite as .
Step 2.1.2.9
Move the negative in front of the fraction.
Step 2.1.3
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 2.1.4
Subtract from both sides of the equation.
Step 2.1.5
Subtract from both sides of the equation.
Step 2.1.6
Divide each term in by and simplify.
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Step 2.1.6.1
Divide each term in by .
Step 2.1.6.2
Simplify the left side.
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Step 2.1.6.2.1
Cancel the common factor of .
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Step 2.1.6.2.1.1
Cancel the common factor.
Step 2.1.6.2.1.2
Divide by .
Step 2.1.6.3
Simplify the right side.
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Step 2.1.6.3.1
Move the negative in front of the fraction.
Step 2.1.7
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 2.1.8
Consolidate the solutions.
Step 2.1.9
Find the domain of .
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Step 2.1.9.1
Set the denominator in equal to to find where the expression is undefined.
Step 2.1.9.2
Solve for .
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Step 2.1.9.2.1
Subtract from both sides of the equation.
Step 2.1.9.2.2
Divide each term in by and simplify.
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Step 2.1.9.2.2.1
Divide each term in by .
Step 2.1.9.2.2.2
Simplify the left side.
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Step 2.1.9.2.2.2.1
Cancel the common factor of .
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Step 2.1.9.2.2.2.1.1
Cancel the common factor.
Step 2.1.9.2.2.2.1.2
Divide by .
Step 2.1.9.2.2.3
Simplify the right side.
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Step 2.1.9.2.2.3.1
Move the negative in front of the fraction.
Step 2.1.9.3
The domain is all values of that make the expression defined.
Step 2.1.10
Use each root to create test intervals.
Step 2.1.11
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 2.1.11.1
Test a value on the interval to see if it makes the inequality true.
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Step 2.1.11.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.1.11.1.2
Replace with in the original inequality.
Step 2.1.11.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 2.1.11.2
Test a value on the interval to see if it makes the inequality true.
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Step 2.1.11.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.1.11.2.2
Replace with in the original inequality.
Step 2.1.11.2.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 2.1.11.3
Test a value on the interval to see if it makes the inequality true.
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Step 2.1.11.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 2.1.11.3.2
Replace with in the original inequality.
Step 2.1.11.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 2.1.11.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 2.1.12
The solution consists of all of the true intervals.
or
or
Step 2.2
Find the intersection of and .
or
or
Step 3
Solve when .
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Step 3.1
Solve for .
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Step 3.1.1
Subtract from both sides of the inequality.
Step 3.1.2
Simplify .
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Step 3.1.2.1
To write as a fraction with a common denominator, multiply by .
Step 3.1.2.2
Combine and .
Step 3.1.2.3
Combine the numerators over the common denominator.
Step 3.1.2.4
Simplify the numerator.
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Step 3.1.2.4.1
Factor out of .
Step 3.1.2.4.2
Add and .
Step 3.1.2.4.3
Add and .
Step 3.1.2.5
Move the negative in front of the fraction.
Step 3.1.3
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 3.1.4
Subtract from both sides of the equation.
Step 3.1.5
Divide each term in by and simplify.
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Step 3.1.5.1
Divide each term in by .
Step 3.1.5.2
Simplify the left side.
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Step 3.1.5.2.1
Cancel the common factor of .
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Step 3.1.5.2.1.1
Cancel the common factor.
Step 3.1.5.2.1.2
Divide by .
Step 3.1.5.3
Simplify the right side.
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Step 3.1.5.3.1
Move the negative in front of the fraction.
Step 3.1.6
Subtract from both sides of the equation.
Step 3.1.7
Divide each term in by and simplify.
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Step 3.1.7.1
Divide each term in by .
Step 3.1.7.2
Simplify the left side.
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Step 3.1.7.2.1
Cancel the common factor of .
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Step 3.1.7.2.1.1
Cancel the common factor.
Step 3.1.7.2.1.2
Divide by .
Step 3.1.7.3
Simplify the right side.
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Step 3.1.7.3.1
Move the negative in front of the fraction.
Step 3.1.8
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 3.1.9
Consolidate the solutions.
Step 3.1.10
Find the domain of .
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Step 3.1.10.1
Set the denominator in equal to to find where the expression is undefined.
Step 3.1.10.2
Solve for .
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Step 3.1.10.2.1
Subtract from both sides of the equation.
Step 3.1.10.2.2
Divide each term in by and simplify.
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Step 3.1.10.2.2.1
Divide each term in by .
Step 3.1.10.2.2.2
Simplify the left side.
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Step 3.1.10.2.2.2.1
Cancel the common factor of .
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Step 3.1.10.2.2.2.1.1
Cancel the common factor.
Step 3.1.10.2.2.2.1.2
Divide by .
Step 3.1.10.2.2.3
Simplify the right side.
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Step 3.1.10.2.2.3.1
Move the negative in front of the fraction.
Step 3.1.10.3
The domain is all values of that make the expression defined.
Step 3.1.11
Use each root to create test intervals.
Step 3.1.12
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
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Step 3.1.12.1
Test a value on the interval to see if it makes the inequality true.
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Step 3.1.12.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.1.12.1.2
Replace with in the original inequality.
Step 3.1.12.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 3.1.12.2
Test a value on the interval to see if it makes the inequality true.
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Step 3.1.12.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.1.12.2.2
Replace with in the original inequality.
Step 3.1.12.2.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 3.1.12.3
Test a value on the interval to see if it makes the inequality true.
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Step 3.1.12.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 3.1.12.3.2
Replace with in the original inequality.
Step 3.1.12.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 3.1.12.4
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
True
False
True
Step 3.1.13
The solution consists of all of the true intervals.
or
or
Step 3.2
Find the intersection of and .
Step 4
Find the union of the solutions.
or
Step 5
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 6