Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈[−23+1,0)∪[23−1,1]
Evaluate
2x33x−1≥1
Find the domain
More Steps
Evaluate
2x3=0
Rewrite the expression
x3=0
The only way a power can not be 0 is when the base not equals 0
x=0
2x33x−1≥1,x=0
Move the expression to the left side
2x33x−1−1≥0
Subtract the terms
More Steps
Evaluate
2x33x−1−1
Reduce fractions to a common denominator
2x33x−1−2x32x3
Write all numerators above the common denominator
2x33x−1−2x3
2x33x−1−2x3≥0
Separate the inequality into 2 possible cases
{3x−1−2x3≥02x3>0{3x−1−2x3≤02x3<0
Solve the inequality
More Steps
Evaluate
3x−1−2x3≥0
Factor the expression
(1−x)(2x2+2x−1)≥0
Separate the inequality into 2 possible cases
{1−x≥02x2+2x−1≥0{1−x≤02x2+2x−1≤0
Solve the inequality
More Steps
Evaluate
1−x≥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
Change the signs on both sides of the inequality and flip the inequality sign
x≤1
{x≤12x2+2x−1≥0{1−x≤02x2+2x−1≤0
Solve the inequality
More Steps
Evaluate
2x2+2x−1≥0
Move the constant to the right side
2x2+2x≥0−(−1)
Add the terms
2x2+2x≥1
Evaluate
x2+x≥21
Add the same value to both sides
x2+x+41≥21+41
Evaluate
x2+x+41≥43
Evaluate
(x+21)2≥43
Take the 2-th root on both sides of the inequality
(x+21)2≥43
Calculate
x+21≥23
Separate the inequality into 2 possible cases
x+21≥23x+21≤−23
Calculate
x≥23−1x+21≤−23
Calculate
x≥23−1x≤−23+1
Find the union
x∈(−∞,−23+1]∪[23−1,+∞)
{x≤1x∈(−∞,−23+1]∪[23−1,+∞){1−x≤02x2+2x−1≤0
Solve the inequality
More Steps
Evaluate
1−x≤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
Change the signs on both sides of the inequality and flip the inequality sign
x≥1
{x≤1x∈(−∞,−23+1]∪[23−1,+∞){x≥12x2+2x−1≤0
Solve the inequality
More Steps
Evaluate
2x2+2x−1≤0
Move the constant to the right side
2x2+2x≤0−(−1)
Add the terms
2x2+2x≤1
Evaluate
x2+x≤21
Add the same value to both sides
x2+x+41≤21+41
Evaluate
x2+x+41≤43
Evaluate
(x+21)2≤43
Take the 2-th root on both sides of the inequality
(x+21)2≤43
Calculate
x+21≤23
Separate the inequality into 2 possible cases
{x+21≤23x+21≥−23
Calculate
{x≤23−1x+21≥−23
Calculate
{x≤23−1x≥−23+1
Find the intersection
−23+1≤x≤23−1
{x≤1x∈(−∞,−23+1]∪[23−1,+∞){x≥1−23+1≤x≤23−1
Find the intersection
x∈(−∞,−23+1]∪[23−1,1]{x≥1−23+1≤x≤23−1
Find the intersection
x∈(−∞,−23+1]∪[23−1,1]x∈∅
Find the union
x∈(−∞,−23+1]∪[23−1,1]
{x∈(−∞,−23+1]∪[23−1,1]2x3>0{3x−1−2x3≤02x3<0
Solve the inequality
More Steps
Evaluate
2x3>0
Rewrite the expression
x3>0
The only way a base raised to an odd power can be greater than 0 is if the base is greater than 0
x>0
{x∈(−∞,−23+1]∪[23−1,1]x>0{3x−1−2x3≤02x3<0
Solve the inequality
More Steps
Evaluate
3x−1−2x3≤0
Factor the expression
(1−x)(2x2+2x−1)≤0
Separate the inequality into 2 possible cases
{1−x≥02x2+2x−1≤0{1−x≤02x2+2x−1≥0
Solve the inequality
More Steps
Evaluate
1−x≥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
Change the signs on both sides of the inequality and flip the inequality sign
x≤1
{x≤12x2+2x−1≤0{1−x≤02x2+2x−1≥0
Solve the inequality
More Steps
Evaluate
2x2+2x−1≤0
Move the constant to the right side
2x2+2x≤0−(−1)
Add the terms
2x2+2x≤1
Evaluate
x2+x≤21
Add the same value to both sides
x2+x+41≤21+41
Evaluate
x2+x+41≤43
Evaluate
(x+21)2≤43
Take the 2-th root on both sides of the inequality
(x+21)2≤43
Calculate
x+21≤23
Separate the inequality into 2 possible cases
{x+21≤23x+21≥−23
Calculate
{x≤23−1x+21≥−23
Calculate
{x≤23−1x≥−23+1
Find the intersection
−23+1≤x≤23−1
{x≤1−23+1≤x≤23−1{1−x≤02x2+2x−1≥0
Solve the inequality
More Steps
Evaluate
1−x≤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
Change the signs on both sides of the inequality and flip the inequality sign
x≥1
{x≤1−23+1≤x≤23−1{x≥12x2+2x−1≥0
Solve the inequality
More Steps
Evaluate
2x2+2x−1≥0
Move the constant to the right side
2x2+2x≥0−(−1)
Add the terms
2x2+2x≥1
Evaluate
x2+x≥21
Add the same value to both sides
x2+x+41≥21+41
Evaluate
x2+x+41≥43
Evaluate
(x+21)2≥43
Take the 2-th root on both sides of the inequality
(x+21)2≥43
Calculate
x+21≥23
Separate the inequality into 2 possible cases
x+21≥23x+21≤−23
Calculate
x≥23−1x+21≤−23
Calculate
x≥23−1x≤−23+1
Find the union
x∈(−∞,−23+1]∪[23−1,+∞)
{x≤1−23+1≤x≤23−1{x≥1x∈(−∞,−23+1]∪[23−1,+∞)
Find the intersection
−23+1≤x≤23−1{x≥1x∈(−∞,−23+1]∪[23−1,+∞)
Find the intersection
−23+1≤x≤23−1x≥1
Find the union
x∈[−23+1,23−1]∪[1,+∞)
{x∈(−∞,−23+1]∪[23−1,1]x>0{x∈[−23+1,23−1]∪[1,+∞)2x3<0
Solve the inequality
More Steps
Evaluate
2x3<0
Rewrite the expression
x3<0
The only way a base raised to an odd power can be less than 0 is if the base is less than 0
x<0
{x∈(−∞,−23+1]∪[23−1,1]x>0{x∈[−23+1,23−1]∪[1,+∞)x<0
Find the intersection
23−1≤x≤1{x∈[−23+1,23−1]∪[1,+∞)x<0
Find the intersection
23−1≤x≤1−23+1≤x<0
Find the union
x∈[−23+1,0)∪[23−1,1]
Check if the solution is in the defined range
x∈[−23+1,0)∪[23−1,1],x=0
Solution
x∈[−23+1,0)∪[23−1,1]
Show Solution
