Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−213+1,2−13+1)∪(213−1,213+1)
Evaluate
x−x3×1<1
Find the domain
x−x3×1<1,x=0
Calculate
x−x3<1
Separate the inequality into 2 possible cases
{x−x3<1x−x3>−1
Solve the inequality for x
More Steps
Evaluate
x−x3<1
Convert the expressions
xx2−3<1
Calculate
xx2−3−1<0
Calculate
More Steps
Calculate
xx2−3−1
Reduce fractions to a common denominator
xx2−3−xx
Write all numerators above the common denominator
xx2−3−x
xx2−3−x<0
Separate the inequality into 2 possible cases
{x2−3−x>0x<0{x2−3−x<0x>0
Solve the inequality
More Steps
Evaluate
x2−3−x>0
Move the constant to the right side
x2−x>0−(−3)
Add the terms
x2−x>3
Add the same value to both sides
x2−x+41>3+41
Evaluate
x2−x+41>413
Evaluate
(x−21)2>413
Take the 2-th root on both sides of the inequality
(x−21)2>413
Calculate
x−21>213
Separate the inequality into 2 possible cases
x−21>213x−21<−213
Calculate
x>213+1x−21<−213
Calculate
x>213+1x<2−13+1
Find the union
x∈(−∞,2−13+1)∪(213+1,+∞)
{x∈(−∞,2−13+1)∪(213+1,+∞)x<0{x2−3−x<0x>0
Solve the inequality
More Steps
Evaluate
x2−3−x<0
Move the constant to the right side
x2−x<0−(−3)
Add the terms
x2−x<3
Add the same value to both sides
x2−x+41<3+41
Evaluate
x2−x+41<413
Evaluate
(x−21)2<413
Take the 2-th root on both sides of the inequality
(x−21)2<413
Calculate
x−21<213
Separate the inequality into 2 possible cases
{x−21<213x−21>−213
Calculate
{x<213+1x−21>−213
Calculate
{x<213+1x>2−13+1
Find the intersection
2−13+1<x<213+1
{x∈(−∞,2−13+1)∪(213+1,+∞)x<0{2−13+1<x<213+1x>0
Find the intersection
x<2−13+1{2−13+1<x<213+1x>0
Find the intersection
x<2−13+10<x<213+1
Find the union
x∈(−∞,2−13+1)∪(0,213+1)
{x∈(−∞,2−13+1)∪(0,213+1)x−x3>−1
Solve the inequality for x
More Steps
Evaluate
x−x3>−1
Convert the expressions
xx2−3>−1
Calculate
xx2−3−(−1)>0
Calculate
More Steps
Calculate
xx2−3−(−1)
Calculate
xx2−3+1
Reduce fractions to a common denominator
xx2−3+xx
Write all numerators above the common denominator
xx2−3+x
xx2−3+x>0
Separate the inequality into 2 possible cases
{x2−3+x>0x>0{x2−3+x<0x<0
Solve the inequality
More Steps
Evaluate
x2−3+x>0
Move the constant to the right side
x2+x>0−(−3)
Add the terms
x2+x>3
Add the same value to both sides
x2+x+41>3+41
Evaluate
x2+x+41>413
Evaluate
(x+21)2>413
Take the 2-th root on both sides of the inequality
(x+21)2>413
Calculate
x+21>213
Separate the inequality into 2 possible cases
x+21>213x+21<−213
Calculate
x>213−1x+21<−213
Calculate
x>213−1x<−213+1
Find the union
x∈(−∞,−213+1)∪(213−1,+∞)
{x∈(−∞,−213+1)∪(213−1,+∞)x>0{x2−3+x<0x<0
Solve the inequality
More Steps
Evaluate
x2−3+x<0
Move the constant to the right side
x2+x<0−(−3)
Add the terms
x2+x<3
Add the same value to both sides
x2+x+41<3+41
Evaluate
x2+x+41<413
Evaluate
(x+21)2<413
Take the 2-th root on both sides of the inequality
(x+21)2<413
Calculate
x+21<213
Separate the inequality into 2 possible cases
{x+21<213x+21>−213
Calculate
{x<213−1x+21>−213
Calculate
{x<213−1x>−213+1
Find the intersection
−213+1<x<213−1
{x∈(−∞,−213+1)∪(213−1,+∞)x>0{−213+1<x<213−1x<0
Find the intersection
x>213−1{−213+1<x<213−1x<0
Find the intersection
x>213−1−213+1<x<0
Find the union
x∈(−213+1,0)∪(213−1,+∞)
⎩⎨⎧x∈(−∞,2−13+1)∪(0,213+1)x∈(−213+1,0)∪(213−1,+∞)
Find the intersection
x∈(−213+1,2−13+1)∪(213−1,213+1)
Check if the solution is in the defined range
x∈(−213+1,2−13+1)∪(213−1,213+1),x=0
Solution
x∈(−213+1,2−13+1)∪(213−1,213+1)
Show Solution
