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Algebra Examples
Step 1
Find all the values where the expression switches from negative to positive by setting each factor equal to and solving.
Step 2
Subtract from both sides of the equation.
Step 3
Set the equal to .
Step 4
Add to both sides of the equation.
Step 5
Add to both sides of the equation.
Step 6
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 7
Step 7.1
Rewrite as .
Step 7.2
Pull terms out from under the radical, assuming positive real numbers.
Step 8
Step 8.1
First, use the positive value of the to find the first solution.
Step 8.2
Next, use the negative value of the to find the second solution.
Step 8.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 9
Solve for each factor to find the values where the absolute value expression goes from negative to positive.
Step 10
Consolidate the solutions.
Step 11
Step 11.1
Set the denominator in equal to to find where the expression is undefined.
Step 11.2
Solve for .
Step 11.2.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 11.2.2
Set equal to and solve for .
Step 11.2.2.1
Set equal to .
Step 11.2.2.2
Solve for .
Step 11.2.2.2.1
Set the equal to .
Step 11.2.2.2.2
Add to both sides of the equation.
Step 11.2.3
Set equal to and solve for .
Step 11.2.3.1
Set equal to .
Step 11.2.3.2
Solve for .
Step 11.2.3.2.1
Add to both sides of the equation.
Step 11.2.3.2.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 11.2.3.2.3
Simplify .
Step 11.2.3.2.3.1
Rewrite as .
Step 11.2.3.2.3.2
Pull terms out from under the radical, assuming positive real numbers.
Step 11.2.3.2.4
The complete solution is the result of both the positive and negative portions of the solution.
Step 11.2.3.2.4.1
First, use the positive value of the to find the first solution.
Step 11.2.3.2.4.2
Next, use the negative value of the to find the second solution.
Step 11.2.3.2.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 11.2.4
The final solution is all the values that make true.
Step 11.3
The domain is all values of that make the expression defined.
Step 12
Use each root to create test intervals.
Step 13
Step 13.1
Test a value on the interval to see if it makes the inequality true.
Step 13.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 13.1.2
Replace with in the original inequality.
Step 13.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 13.2
Test a value on the interval to see if it makes the inequality true.
Step 13.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 13.2.2
Replace with in the original inequality.
Step 13.2.3
The left side is greater than the right side , which means that the given statement is false.
False
False
Step 13.3
Test a value on the interval to see if it makes the inequality true.
Step 13.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 13.3.2
Replace with in the original inequality.
Step 13.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 13.4
Test a value on the interval to see if it makes the inequality true.
Step 13.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 13.4.2
Replace with in the original inequality.
Step 13.4.3
The left side is greater than the right side , which means that the given statement is false.
False
False
Step 13.5
Test a value on the interval to see if it makes the inequality true.
Step 13.5.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 13.5.2
Replace with in the original inequality.
Step 13.5.3
The left side is greater than the right side , which means that the given statement is false.
False
False
Step 13.6
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
False
True
False
True
False
False
Step 14
The solution consists of all of the true intervals.
or
Step 15
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 16