Question
Solve the inequality
x∈(−32,0)∪(0,52)
Evaluate
xx−2>4
Find the domain
xx−2>4,x=0
Separate the inequality into 2 possible cases
xx−2>4xx−2<−4
Solve the inequality for x
More Steps

Evaluate
xx−2>4
Calculate
xx−2−4>0
Calculate
More Steps

Calculate
xx−2−4
Reduce fractions to a common denominator
xx−2−x4x
Write all numerators above the common denominator
xx−2−4x
Subtract the terms
x−3x−2
Use b−a=−ba=−ba to rewrite the fraction
−x3x+2
−x3x+2>0
Change the signs on both sides of the inequality and flip the inequality sign
x3x+2<0
Separate the inequality into 2 possible cases
{3x+2>0x<0{3x+2<0x>0
Solve the inequality
More Steps

Evaluate
3x+2>0
Move the constant to the right side
3x>0−2
Removing 0 doesn't change the value,so remove it from the expression
3x>−2
Divide both sides
33x>3−2
Divide the numbers
x>3−2
Use b−a=−ba=−ba to rewrite the fraction
x>−32
{x>−32x<0{3x+2<0x>0
Solve the inequality
More Steps

Evaluate
3x+2<0
Move the constant to the right side
3x<0−2
Removing 0 doesn't change the value,so remove it from the expression
3x<−2
Divide both sides
33x<3−2
Divide the numbers
x<3−2
Use b−a=−ba=−ba to rewrite the fraction
x<−32
{x>−32x<0{x<−32x>0
Find the intersection
−32<x<0{x<−32x>0
Find the intersection
−32<x<0x∈∅
Find the union
−32<x<0
−32<x<0xx−2<−4
Solve the inequality for x
More Steps

Evaluate
xx−2<−4
Calculate
xx−2−(−4)<0
Calculate
More Steps

Calculate
xx−2−(−4)
Calculate
xx−2+4
Reduce fractions to a common denominator
xx−2+x4x
Write all numerators above the common denominator
xx−2+4x
Add the terms
x5x−2
x5x−2<0
Separate the inequality into 2 possible cases
{5x−2>0x<0{5x−2<0x>0
Solve the inequality
More Steps

Evaluate
5x−2>0
Move the constant to the right side
5x>0+2
Removing 0 doesn't change the value,so remove it from the expression
5x>2
Divide both sides
55x>52
Divide the numbers
x>52
{x>52x<0{5x−2<0x>0
Solve the inequality
More Steps

Evaluate
5x−2<0
Move the constant to the right side
5x<0+2
Removing 0 doesn't change the value,so remove it from the expression
5x<2
Divide both sides
55x<52
Divide the numbers
x<52
{x>52x<0{x<52x>0
Find the intersection
x∈∅{x<52x>0
Find the intersection
x∈∅0<x<52
Find the union
0<x<52
−32<x<00<x<52
Find the union
x∈(−32,0)∪(0,52)
Check if the solution is in the defined range
x∈(−32,0)∪(0,52),x=0
Solution
x∈(−32,0)∪(0,52)
Show Solution
