Finite Math Examples

Convert to Interval Notation |2x-4|<|x-1|
Step 1
Replace with in .
Step 2
Rewrite the absolute value equation as four equations without absolute value bars.
Step 3
After simplifying, there are only two unique equations to be solved.
Step 4
Solve for .
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Step 4.1
Move all terms containing to the left side of the equation.
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Step 4.1.1
Subtract from both sides of the equation.
Step 4.1.2
Subtract from .
Step 4.2
Move all terms not containing to the right side of the equation.
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Step 4.2.1
Add to both sides of the equation.
Step 4.2.2
Add and .
Step 5
Solve for .
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Step 5.1
Simplify .
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Step 5.1.1
Rewrite.
Step 5.1.2
Simplify by adding zeros.
Step 5.1.3
Apply the distributive property.
Step 5.1.4
Multiply by .
Step 5.2
Move all terms containing to the left side of the equation.
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Step 5.2.1
Add to both sides of the equation.
Step 5.2.2
Add and .
Step 5.3
Move all terms not containing to the right side of the equation.
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Step 5.3.1
Add to both sides of the equation.
Step 5.3.2
Add and .
Step 5.4
Divide each term in by and simplify.
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Step 5.4.1
Divide each term in by .
Step 5.4.2
Simplify the left side.
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Step 5.4.2.1
Cancel the common factor of .
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Step 5.4.2.1.1
Cancel the common factor.
Step 5.4.2.1.2
Divide by .
Step 6
List all of the solutions.
Step 7
Use each root to create test intervals.
Step 8
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 8.1
Test a value on the interval to see if it makes the inequality true.
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Step 8.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.1.2
Replace with in the original inequality.
Step 8.1.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 8.2
Test a value on the interval to see if it makes the inequality true.
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Step 8.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.2.2
Replace with in the original inequality.
Step 8.2.3
The left side is less than the right side , which means that the given statement is always true.
True
True
Step 8.3
Test a value on the interval to see if it makes the inequality true.
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Step 8.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.3.2
Replace with in the original inequality.
Step 8.3.3
The left side is not less than the right side , which means that the given statement is false.
False
False
Step 8.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 9
The solution consists of all of the true intervals.
Step 10
Convert the inequality to interval notation.
Step 11