Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,−1]∪[0,+∞)
Evaluate
x3−1≥1−x
Rearrange the terms
x3−1+x≥1
Separate the inequality into 2 possible cases
x3−1+x≥1,x3−1≥0−(x3−1)+x≥1,x3−1<0
Evaluate
More Steps
Evaluate
x3−1+x≥1
Move the expression to the left side
x3−1+x−1≥0
Calculate
x3−2+x≥0
Factor the expression
(x−1)(x2+x+2)≥0
Separate the inequality into 2 possible cases
{x−1≥0x2+x+2≥0{x−1≤0x2+x+2≤0
Solve the inequality
More Steps
Evaluate
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≥1x2+x+2≥0{x−1≤0x2+x+2≤0
Solve the inequality
More Steps
Evaluate
x2+x+2≥0
Move the constant to the right side
x2+x≥0−2
Add the terms
x2+x≥−2
Add the same value to both sides
x2+x+41≥−2+41
Evaluate
x2+x+41≥−47
Evaluate
(x+21)2≥−47
Calculate
x∈R
{x≥1x∈R{x−1≤0x2+x+2≤0
Solve the inequality
More Steps
Evaluate
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≥1x∈R{x≤1x2+x+2≤0
Solve the inequality
More Steps
Evaluate
x2+x+2≤0
Move the constant to the right side
x2+x≤0−2
Add the terms
x2+x≤−2
Add the same value to both sides
x2+x+41≤−2+41
Evaluate
x2+x+41≤−47
Evaluate
(x+21)2≤−47
Calculate
x∈/R
{x≥1x∈R{x≤1x∈/R
Find the intersection
x≥1{x≤1x∈/R
Find the intersection
x≥1x∈/R
Find the union
x≥1
x≥1,x3−1≥0−(x3−1)+x≥1,x3−1<0
Evaluate
More Steps
Evaluate
x3−1≥0
Move the constant to the right side
x3≥1
Take the 3-th root on both sides of the equation
3x3≥31
Calculate
x≥31
Simplify the root
x≥1
x≥1,x≥1−(x3−1)+x≥1,x3−1<0
Evaluate
More Steps
Evaluate
−(x3−1)+x≥1
Remove the parentheses
−x3+1+x≥1
Move the expression to the left side
−x3+1+x−1≥0
Calculate
−x3+0+x≥0
Evaluate
−x3+x≥0
Factor the expression
x(−x2+1)≥0
Separate the inequality into 2 possible cases
{x≥0−x2+1≥0{x≤0−x2+1≤0
Solve the inequality
More Steps
Evaluate
−x2+1≥0
Move the constant to the right side
−x2≥−1
Change the signs on both sides of the inequality and flip the inequality sign
x2≤1
Take the 2-th root on both sides of the inequality
x2≤1
Calculate
∣x∣≤1
Separate the inequality into 2 possible cases
{x≤1x≥−1
Find the intersection
−1≤x≤1
{x≥0−1≤x≤1{x≤0−x2+1≤0
Solve the inequality
More Steps
Evaluate
−x2+1≤0
Move the constant to the right side
−x2≤−1
Change the signs on both sides of the inequality and flip the inequality sign
x2≥1
Take the 2-th root on both sides of the inequality
x2≥1
Calculate
∣x∣≥1
Separate the inequality into 2 possible cases
x≥1x≤−1
Find the union
x∈(−∞,−1]∪[1,+∞)
{x≥0−1≤x≤1{x≤0x∈(−∞,−1]∪[1,+∞)
Find the intersection
0≤x≤1{x≤0x∈(−∞,−1]∪[1,+∞)
Find the intersection
0≤x≤1x≤−1
Find the union
x∈(−∞,−1]∪[0,1]
x≥1,x≥1x∈(−∞,−1]∪[0,1],x3−1<0
Evaluate
More Steps
Evaluate
x3−1<0
Move the constant to the right side
x3<1
Take the 3-th root on both sides of the equation
3x3<31
Calculate
x<31
Simplify the root
x<1
x≥1,x≥1x∈(−∞,−1]∪[0,1],x<1
Find the intersection
x≥1x∈(−∞,−1]∪[0,1],x<1
Find the intersection
x≥1x∈(−∞,−1]∪[0,1)
Solution
x∈(−∞,−1]∪[0,+∞)
Show Solution
