x1-x>1

Move 1 to the left side of the equation by subtracting it from both sides.

x1-x-1>0

To write -1 as a fraction with a common denominator, multiply by 1-x1-x.

x1-x-1⋅1-x1-x>0

Simplify terms.

Combine -1 and 1-x1-x.

x1-x+-(1-x)1-x>0

Combine the numerators over the common denominator.

x-(1-x)1-x>0

x-(1-x)1-x>0

Simplify the numerator.

Apply the distributive property.

x-1⋅1–x1-x>0

Multiply -1 by 1.

x-1–x1-x>0

Multiply –x.

Multiply -1 by -1.

x-1+1×1-x>0

Multiply x by 1.

x-1+x1-x>0

x-1+x1-x>0

Add x and x.

2x-11-x>0

2x-11-x>0

2x-11-x>0

Find all the values where the expression switches from negative to positive by setting each factor equal to 0 and solving.

2x-1=0

1-x=0

Add 1 to both sides of the equation.

2x=1

Divide each term in 2x=1 by 2.

2×2=12

Cancel the common factor of 2.

Cancel the common factor.

2×2=12

Divide x by 1.

x=12

x=12

x=12

Subtract 1 from both sides of the equation.

-x=-1

Multiply each term in -x=-1 by -1.

(-x)⋅-1=(-1)⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x=(-1)⋅-1

Multiply x by 1.

x=(-1)⋅-1

x=(-1)⋅-1

Multiply -1 by -1.

x=1

x=1

Solve for each factor to find the values where the absolute value expression goes from negative to positive.

x=12

x=1

Consolidate the solutions.

x=12,1

Set the denominator in 2x-11-x equal to 0 to find where the expression is undefined.

1-x=0

Solve for x.

Subtract 1 from both sides of the equation.

-x=-1

Multiply each term in -x=-1 by -1

Multiply each term in -x=-1 by -1.

(-x)⋅-1=(-1)⋅-1

Multiply (-x)⋅-1.

Multiply -1 by -1.

1x=(-1)⋅-1

Multiply x by 1.

x=(-1)⋅-1

x=(-1)⋅-1

Multiply -1 by -1.

x=1

x=1

x=1

The domain is all values of x that make the expression defined.

(-∞,1)∪(1,∞)

(-∞,1)∪(1,∞)

Use each root to create test intervals.

x<12

12<x<1

x>1

Test a value on the interval x<12 to see if it makes the inequality true.

Choose a value on the interval x<12 and see if this value makes the original inequality true.

x=0

Replace x with 0 in the original inequality.

01-(0)>1

The left side 0 is not greater than the right side 1, which means that the given statement is false.

False

False

Test a value on the interval 12<x<1 to see if it makes the inequality true.

Choose a value on the interval 12<x<1 and see if this value makes the original inequality true.

x=0.75

Replace x with 0.75 in the original inequality.

0.751-(0.75)>1

The left side 3 is greater than the right side 1, which means that the given statement is always true.

True

True

Test a value on the interval x>1 to see if it makes the inequality true.

Choose a value on the interval x>1 and see if this value makes the original inequality true.

x=4

Replace x with 4 in the original inequality.

41-(4)>1

The left side -1.3‾ is not greater than the right side 1, which means that the given statement is false.

False

False

Compare the intervals to determine which ones satisfy the original inequality.

x<12 False

12<x<1 True

x>1 False

x<12 False

12<x<1 True

x>1 False

The solution consists of all of the true intervals.

12<x<1

The result can be shown in multiple forms.

Inequality Form:

12<x<1

Interval Notation:

(12,1)

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