Pre-Algebra Examples

Graph |x/3-x/4|-1<0
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
Simplify .
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Step 1.2.1.1
To write as a fraction with a common denominator, multiply by .
Step 1.2.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.2.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.2.1.3.1
Multiply by .
Step 1.2.1.3.2
Multiply by .
Step 1.2.1.3.3
Multiply by .
Step 1.2.1.3.4
Multiply by .
Step 1.2.1.4
Combine the numerators over the common denominator.
Step 1.2.1.5
Simplify the numerator.
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Step 1.2.1.5.1
Factor out of .
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Step 1.2.1.5.1.1
Factor out of .
Step 1.2.1.5.1.2
Factor out of .
Step 1.2.1.5.1.3
Factor out of .
Step 1.2.1.5.2
Multiply by .
Step 1.2.1.5.3
Subtract from .
Step 1.2.1.6
Multiply by .
Step 1.2.2
Multiply both sides by .
Step 1.2.3
Simplify.
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Step 1.2.3.1
Simplify the left side.
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Step 1.2.3.1.1
Cancel the common factor of .
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Step 1.2.3.1.1.1
Cancel the common factor.
Step 1.2.3.1.1.2
Rewrite the expression.
Step 1.2.3.2
Simplify the right side.
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Step 1.2.3.2.1
Multiply by .
Step 1.3
In the piece where is non-negative, remove the absolute value.
Step 1.4
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 1.5
Solve the inequality.
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Step 1.5.1
Simplify .
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Step 1.5.1.1
To write as a fraction with a common denominator, multiply by .
Step 1.5.1.2
To write as a fraction with a common denominator, multiply by .
Step 1.5.1.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.5.1.3.1
Multiply by .
Step 1.5.1.3.2
Multiply by .
Step 1.5.1.3.3
Multiply by .
Step 1.5.1.3.4
Multiply by .
Step 1.5.1.4
Combine the numerators over the common denominator.
Step 1.5.1.5
Simplify the numerator.
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Step 1.5.1.5.1
Factor out of .
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Step 1.5.1.5.1.1
Factor out of .
Step 1.5.1.5.1.2
Factor out of .
Step 1.5.1.5.1.3
Factor out of .
Step 1.5.1.5.2
Multiply by .
Step 1.5.1.5.3
Subtract from .
Step 1.5.1.6
Multiply by .
Step 1.5.2
Multiply both sides by .
Step 1.5.3
Simplify.
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Step 1.5.3.1
Simplify the left side.
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Step 1.5.3.1.1
Cancel the common factor of .
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Step 1.5.3.1.1.1
Cancel the common factor.
Step 1.5.3.1.1.2
Rewrite the expression.
Step 1.5.3.2
Simplify the right side.
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Step 1.5.3.2.1
Multiply by .
Step 1.6
In the piece where is negative, remove the absolute value and multiply by .
Step 1.7
Write as a piecewise.
Step 1.8
Simplify .
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Step 1.8.1
To write as a fraction with a common denominator, multiply by .
Step 1.8.2
To write as a fraction with a common denominator, multiply by .
Step 1.8.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.8.3.1
Multiply by .
Step 1.8.3.2
Multiply by .
Step 1.8.3.3
Multiply by .
Step 1.8.3.4
Multiply by .
Step 1.8.4
Combine the numerators over the common denominator.
Step 1.8.5
Simplify each term.
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Step 1.8.5.1
Simplify the numerator.
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Step 1.8.5.1.1
Factor out of .
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Step 1.8.5.1.1.1
Factor out of .
Step 1.8.5.1.1.2
Factor out of .
Step 1.8.5.1.1.3
Factor out of .
Step 1.8.5.1.2
Multiply by .
Step 1.8.5.1.3
Subtract from .
Step 1.8.5.2
Multiply by .
Step 1.9
Simplify each term.
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Step 1.9.1
To write as a fraction with a common denominator, multiply by .
Step 1.9.2
To write as a fraction with a common denominator, multiply by .
Step 1.9.3
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 1.9.3.1
Multiply by .
Step 1.9.3.2
Multiply by .
Step 1.9.3.3
Multiply by .
Step 1.9.3.4
Multiply by .
Step 1.9.4
Combine the numerators over the common denominator.
Step 1.9.5
Simplify the numerator.
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Step 1.9.5.1
Factor out of .
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Step 1.9.5.1.1
Factor out of .
Step 1.9.5.1.2
Factor out of .
Step 1.9.5.1.3
Factor out of .
Step 1.9.5.2
Multiply by .
Step 1.9.5.3
Subtract from .
Step 1.9.6
Multiply by .
Step 2
Solve when .
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Step 2.1
Solve for .
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Step 2.1.1
Add to both sides of the inequality.
Step 2.1.2
Multiply both sides by .
Step 2.1.3
Simplify.
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Step 2.1.3.1
Simplify the left side.
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Step 2.1.3.1.1
Cancel the common factor of .
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Step 2.1.3.1.1.1
Cancel the common factor.
Step 2.1.3.1.1.2
Rewrite the expression.
Step 2.1.3.2
Simplify the right side.
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Step 2.1.3.2.1
Multiply by .
Step 2.2
Find the intersection of and .
Step 3
Solve when .
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Step 3.1
Solve for .
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Step 3.1.1
Add to both sides of the inequality.
Step 3.1.2
Divide each term in by and simplify.
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Step 3.1.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 3.1.2.2
Simplify the left side.
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Step 3.1.2.2.1
Dividing two negative values results in a positive value.
Step 3.1.2.2.2
Divide by .
Step 3.1.2.3
Simplify the right side.
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Step 3.1.2.3.1
Divide by .
Step 3.1.3
Multiply both sides by .
Step 3.1.4
Simplify.
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Step 3.1.4.1
Simplify the left side.
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Step 3.1.4.1.1
Cancel the common factor of .
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Step 3.1.4.1.1.1
Cancel the common factor.
Step 3.1.4.1.1.2
Rewrite the expression.
Step 3.1.4.2
Simplify the right side.
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Step 3.1.4.2.1
Multiply by .
Step 3.2
Find the intersection of and .
Step 4
Find the union of the solutions.
Step 5