Pre-Algebra Examples

Solve for x x/(1-x)>1
Step 1
Subtract from both sides of the inequality.
Step 2
Simplify .
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Step 2.1
To write as a fraction with a common denominator, multiply by .
Step 2.2
Combine and .
Step 2.3
Combine the numerators over the common denominator.
Step 2.4
Simplify the numerator.
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Step 2.4.1
Apply the distributive property.
Step 2.4.2
Multiply by .
Step 2.4.3
Multiply .
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Step 2.4.3.1
Multiply by .
Step 2.4.3.2
Multiply by .
Step 2.4.4
Add and .
Step 3
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 4
Add to both sides of the equation.
Step 5
Divide each term in by and simplify.
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Step 5.1
Divide each term in by .
Step 5.2
Simplify the left side.
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Step 5.2.1
Cancel the common factor of .
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Step 5.2.1.1
Cancel the common factor.
Step 5.2.1.2
Divide by .
Step 6
Subtract from both sides of the equation.
Step 7
Divide each term in by and simplify.
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Step 7.1
Divide each term in by .
Step 7.2
Simplify the left side.
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Step 7.2.1
Dividing two negative values results in a positive value.
Step 7.2.2
Divide by .
Step 7.3
Simplify the right side.
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Step 7.3.1
Divide by .
Step 8
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 9
Consolidate the solutions.
Step 10
Find the domain of .
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Step 10.1
Set the denominator in equal to to find where the expression is undefined.
Step 10.2
Solve for .
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Step 10.2.1
Subtract from both sides of the equation.
Step 10.2.2
Divide each term in by and simplify.
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Step 10.2.2.1
Divide each term in by .
Step 10.2.2.2
Simplify the left side.
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Step 10.2.2.2.1
Dividing two negative values results in a positive value.
Step 10.2.2.2.2
Divide by .
Step 10.2.2.3
Simplify the right side.
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Step 10.2.2.3.1
Divide by .
Step 10.3
The domain is all values of that make the expression defined.
Step 11
Use each root to create test intervals.
Step 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 12.1
Test a value on the interval to see if it makes the inequality true.
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Step 12.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.1.2
Replace with in the original inequality.
Step 12.1.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 12.2
Test a value on the interval to see if it makes the inequality true.
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Step 12.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.2.2
Replace with in the original inequality.
Step 12.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 12.3
Test a value on the interval to see if it makes the inequality true.
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Step 12.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 12.3.2
Replace with in the original inequality.
Step 12.3.3
The left side is not greater than the right side , which means that the given statement is false.
False
False
Step 12.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 13
The solution consists of all of the true intervals.
Step 14
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 15