Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,0)∪(0,21)
Evaluate
∣x−1∣x2<∣x∣
Find the domain
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Evaluate
∣x−1∣=0
Rewrite the expression
x−1=0
Move the constant to the right side
x=0+1
Removing 0 doesn't change the value,so remove it from the expression
x=1
∣x−1∣x2<∣x∣,x=1
Move the expression to the left side
∣x−1∣x2−∣x∣<0
Subtract the terms
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Evaluate
∣x−1∣x2−∣x∣
Reduce fractions to a common denominator
∣x−1∣x2−∣x−1∣∣x∣∣x−1∣
Write all numerators above the common denominator
∣x−1∣x2−∣x∣∣x−1∣
Multiply the terms
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Evaluate
x(x−1)
Apply the distributive property
x×x−x×1
Multiply the terms
x2−x×1
Any expression multiplied by 1 remains the same
x2−x
∣x−1∣x2−x2−x
∣x−1∣x2−x2−x<0
Separate the inequality into 2 possible cases
{x2−x2−x>0∣x−1∣<0{x2−x2−x<0∣x−1∣>0
Solve the inequality
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Evaluate
x2−x2−x>0
Separate the inequality into 2 possible cases
x2−(x2−x)>0,x2−x≥0x2−(−(x2−x))>0,x2−x<0
Evaluate
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Evaluate
x2−(x2−x)>0
Remove the parentheses
x2−x2+x>0
Simplify the expression
x>0
x>0,x2−x≥0x2−(−(x2−x))>0,x2−x<0
Evaluate
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Evaluate
x2−x≥0
Add the same value to both sides
x2−x+41≥41
Evaluate
(x−21)2≥41
Take the 2-th root on both sides of the inequality
(x−21)2≥41
Calculate
x−21≥21
Separate the inequality into 2 possible cases
x−21≥21x−21≤−21
Calculate
x≥1x−21≤−21
Cancel equal terms on both sides of the expression
x≥1x≤0
Find the union
x∈(−∞,0]∪[1,+∞)
x>0,x∈(−∞,0]∪[1,+∞)x2−(−(x2−x))>0,x2−x<0
Evaluate
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Evaluate
x2−(−(x2−x))>0
Remove the parentheses
x2+x2−x>0
Simplify the expression
2x2−x>0
Evaluate
x2−21x>0
Add the same value to both sides
x2−21x+161>161
Evaluate
(x−41)2>161
Take the 2-th root on both sides of the inequality
(x−41)2>161
Calculate
x−41>41
Separate the inequality into 2 possible cases
x−41>41x−41<−41
Calculate
x>21x−41<−41
Cancel equal terms on both sides of the expression
x>21x<0
Find the union
x∈(−∞,0)∪(21,+∞)
x>0,x∈(−∞,0]∪[1,+∞)x∈(−∞,0)∪(21,+∞),x2−x<0
Evaluate
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Evaluate
x2−x<0
Add the same value to both sides
x2−x+41<41
Evaluate
(x−21)2<41
Take the 2-th root on both sides of the inequality
(x−21)2<41
Calculate
x−21<21
Separate the inequality into 2 possible cases
{x−21<21x−21>−21
Calculate
{x<1x−21>−21
Cancel equal terms on both sides of the expression
{x<1x>0
Find the intersection
0<x<1
x>0,x∈(−∞,0]∪[1,+∞)x∈(−∞,0)∪(21,+∞),0<x<1
Find the intersection
x≥1x∈(−∞,0)∪(21,+∞),0<x<1
Find the intersection
x≥121<x<1
Find the union
x>21
{x>21∣x−1∣<0{x2−x2−x<0∣x−1∣>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is false for any value of x
{x>21x∈/R{x2−x2−x<0∣x−1∣>0
Solve the inequality
More Steps
Evaluate
x2−x2−x<0
Separate the inequality into 2 possible cases
x2−(x2−x)<0,x2−x≥0x2−(−(x2−x))<0,x2−x<0
Evaluate
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Evaluate
x2−(x2−x)<0
Remove the parentheses
x2−x2+x<0
Simplify the expression
x<0
x<0,x2−x≥0x2−(−(x2−x))<0,x2−x<0
Evaluate
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Evaluate
x2−x≥0
Add the same value to both sides
x2−x+41≥41
Evaluate
(x−21)2≥41
Take the 2-th root on both sides of the inequality
(x−21)2≥41
Calculate
x−21≥21
Separate the inequality into 2 possible cases
x−21≥21x−21≤−21
Calculate
x≥1x−21≤−21
Cancel equal terms on both sides of the expression
x≥1x≤0
Find the union
x∈(−∞,0]∪[1,+∞)
x<0,x∈(−∞,0]∪[1,+∞)x2−(−(x2−x))<0,x2−x<0
Evaluate
More Steps
Evaluate
x2−(−(x2−x))<0
Remove the parentheses
x2+x2−x<0
Simplify the expression
2x2−x<0
Evaluate
x2−21x<0
Add the same value to both sides
x2−21x+161<161
Evaluate
(x−41)2<161
Take the 2-th root on both sides of the inequality
(x−41)2<161
Calculate
x−41<41
Separate the inequality into 2 possible cases
{x−41<41x−41>−41
Calculate
{x<21x−41>−41
Cancel equal terms on both sides of the expression
{x<21x>0
Find the intersection
0<x<21
x<0,x∈(−∞,0]∪[1,+∞)0<x<21,x2−x<0
Evaluate
More Steps
Evaluate
x2−x<0
Add the same value to both sides
x2−x+41<41
Evaluate
(x−21)2<41
Take the 2-th root on both sides of the inequality
(x−21)2<41
Calculate
x−21<21
Separate the inequality into 2 possible cases
{x−21<21x−21>−21
Calculate
{x<1x−21>−21
Cancel equal terms on both sides of the expression
{x<1x>0
Find the intersection
0<x<1
x<0,x∈(−∞,0]∪[1,+∞)0<x<21,0<x<1
Find the intersection
x<00<x<21,0<x<1
Find the intersection
x<00<x<21
Find the union
x∈(−∞,0)∪(0,21)
{x>21x∈/R{x∈(−∞,0)∪(0,21)∣x−1∣>0
Solve the inequality
More Steps
Evaluate
∣x−1∣>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x,except when ∣x−1∣=0
∣x−1∣=0
Rewrite the expression
x−1=0
Move the constant to the right-hand side and change its sign
x=0+1
Removing 0 doesn't change the value,so remove it from the expression
x=1
Exclude the impossible values of x
x=1
{x>21x∈/R{x∈(−∞,0)∪(0,21)x=1
Find the intersection
x∈/R{x∈(−∞,0)∪(0,21)x=1
Find the intersection
x∈/Rx∈(−∞,0)∪(0,21)
Find the union
x∈(−∞,0)∪(0,21)
Check if the solution is in the defined range
x∈(−∞,0)∪(0,21),x=1
Solution
x∈(−∞,0)∪(0,21)
Show Solution
