Algebra Examples

Solve the Inequality for x x+4<5/x
Step 1
Multiply both sides by .
Step 2
Simplify.
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Step 2.1
Simplify the left side.
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Step 2.1.1
Simplify .
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Step 2.1.1.1
Apply the distributive property.
Step 2.1.1.2
Multiply by .
Step 2.2
Simplify the right side.
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Step 2.2.1
Cancel the common factor of .
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Step 2.2.1.1
Cancel the common factor.
Step 2.2.1.2
Rewrite the expression.
Step 3
Solve for .
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Step 3.1
Subtract from both sides of the equation.
Step 3.2
Factor using the AC method.
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Step 3.2.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 3.2.2
Write the factored form using these integers.
Step 3.3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 3.4
Set equal to and solve for .
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Step 3.4.1
Set equal to .
Step 3.4.2
Add to both sides of the equation.
Step 3.5
Set equal to and solve for .
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Step 3.5.1
Set equal to .
Step 3.5.2
Subtract from both sides of the equation.
Step 3.6
The final solution is all the values that make true.
Step 4
Find the domain of .
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Step 4.1
Set the denominator in equal to to find where the expression is undefined.
Step 4.2
The domain is all values of that make the expression defined.
Step 5
Use each root to create test intervals.
Step 6
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 6.1
Test a value on the interval to see if it makes the inequality true.
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Step 6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.1.2
Replace with in the original inequality.
Step 6.1.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 6.2
Test a value on the interval to see if it makes the inequality true.
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Step 6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.2.2
Replace with in the original inequality.
Step 6.2.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 6.3
Test a value on the interval to see if it makes the inequality true.
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Step 6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.3.2
Replace with in the original inequality.
Step 6.3.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 6.4
Test a value on the interval to see if it makes the inequality true.
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Step 6.4.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 6.4.2
Replace with in the original inequality.
Step 6.4.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 6.5
Compare the intervals to determine which ones satisfy the original inequality.
True
False
True
False
True
False
True
False
Step 7
The solution consists of all of the true intervals.
or
Step 8
The result can be shown in multiple forms.
Inequality Form:
Interval Notation:
Step 9