Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,−1.111816]∪[0.855758,+∞)
Evaluate
x−1−x2≤x2−3x4
Rearrange the terms
x−1−x2−x2−3x4≤0
Separate the inequality into 4 possible cases
x−1−x2−(x2−3x4)≤0,x−1−x2≥0,x2−3x4≥0x−1−x2−(−(x2−3x4))≤0,x−1−x2≥0,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2−(x2−3x4)≤0
Remove the parentheses
x−1−x2−x2+3x4≤0
Simplify the expression
x−1−2x2+3x4≤0
Rewrite the expression
x−1−2x2+3x4=0
Find the critical values by solving the corresponding equation
x≈0.855758x≈−1.111816
Determine the test intervals using the critical values
x<−1.111816−1.111816<x<0.855758x>0.855758
Choose a value form each interval
x1=−2x2=0x3=2
To determine if x<−1.111816 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
More Steps
Evaluate
−2−1−2(−2)2+3(−2)4≤0
Simplify
37≤0
Check the inequality
false
x<−1.111816 is not a solutionx2=0x3=2
To determine if −1.111816<x<0.855758 is the solution to the inequality,test if the chosen value x=0 satisfies the initial inequality
More Steps
Evaluate
0−1−2×02+3×04≤0
Simplify
−1≤0
Check the inequality
true
x<−1.111816 is not a solution−1.111816<x<0.855758 is the solutionx3=2
To determine if x>0.855758 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
2−1−2×22+3×24≤0
Simplify
41≤0
Check the inequality
false
x<−1.111816 is not a solution−1.111816<x<0.855758 is the solutionx>0.855758 is not a solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
−1.111816≤x≤0.855758 is the solution
The final solution of the original inequality is −1.111816≤x≤0.855758
−1.111816≤x≤0.855758
−1.111816≤x≤0.855758,x−1−x2≥0,x2−3x4≥0x−1−x2−(−(x2−3x4))≤0,x−1−x2≥0,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2≥0
Move the constant to the right side
x−x2≥0−(−1)
Add the terms
x−x2≥1
Evaluate
x2−x≤−1
Add the same value to both sides
x2−x+41≤−1+41
Evaluate
x2−x+41≤−43
Evaluate
(x−21)2≤−43
Calculate
x∈/R
−1.111816≤x≤0.855758,x∈/R,x2−3x4≥0x−1−x2−(−(x2−3x4))≤0,x−1−x2≥0,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x2−3x4≥0
Factor the expression
x2(1−3x2)≥0
Separate the inequality into 2 possible cases
{x2≥01−3x2≥0{x2≤01−3x2≤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
{x∈R1−3x2≥0{x2≤01−3x2≤0
Solve the inequality
More Steps
Evaluate
1−3x2≥0
Rewrite the expression
−3x2≥−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2≤1
Divide both sides
33x2≤31
Divide the numbers
x2≤31
Take the 2-th root on both sides of the inequality
x2≤31
Calculate
∣x∣≤33
Separate the inequality into 2 possible cases
{x≤33x≥−33
Find the intersection
−33≤x≤33
{x∈R−33≤x≤33{x2≤01−3x2≤0
Solve the inequality
More Steps
Evaluate
x2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true only when x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
{x∈R−33≤x≤33{x=01−3x2≤0
Solve the inequality
More Steps
Evaluate
1−3x2≤0
Rewrite the expression
−3x2≤−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2≥1
Divide both sides
33x2≥31
Divide the numbers
x2≥31
Take the 2-th root on both sides of the inequality
x2≥31
Calculate
∣x∣≥33
Separate the inequality into 2 possible cases
x≥33x≤−33
Find the union
x∈(−∞,−33]∪[33,+∞)
{x∈R−33≤x≤33{x=0x∈(−∞,−33]∪[33,+∞)
Find the intersection
−33≤x≤33{x=0x∈(−∞,−33]∪[33,+∞)
Find the intersection
−33≤x≤33x∈∅
Find the union
−33≤x≤33
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x−1−x2−(−(x2−3x4))≤0,x−1−x2≥0,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2−(−(x2−3x4))≤0
Remove the parentheses
x−1−x2+x2−3x4≤0
Simplify the expression
x−1−3x4≤0
Rewrite the expression
x−1−3x4=0
Find the critical values by solving the corresponding equation
x∈/R
There are no key numbers,so choose any value to test,for example x=0
x=0
To determine if the solution to the inequality are all real numbers,test if the chosen value satisfies the initial inequality
More Steps
Evaluate
0−1−3×04≤0
Simplify
−1≤0
Check the inequality
true
x∈R
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x−1−x2≥0,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2≥0
Move the constant to the right side
x−x2≥0−(−1)
Add the terms
x−x2≥1
Evaluate
x2−x≤−1
Add the same value to both sides
x2−x+41≤−1+41
Evaluate
x2−x+41≤−43
Evaluate
(x−21)2≤−43
Calculate
x∈/R
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x2−3x4<0−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x2−3x4<0
Factor the expression
x2(1−3x2)<0
Separate the inequality into 2 possible cases
{x2>01−3x2<0{x2<01−3x2>0
Solve the inequality
More Steps
Evaluate
x2>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 x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
Exclude the impossible values of x
x=0
{x=01−3x2<0{x2<01−3x2>0
Solve the inequality
More Steps
Evaluate
1−3x2<0
Rewrite the expression
−3x2<−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2>1
Divide both sides
33x2>31
Divide the numbers
x2>31
Take the 2-th root on both sides of the inequality
x2>31
Calculate
∣x∣>33
Separate the inequality into 2 possible cases
x>33x<−33
Find the union
x∈(−∞,−33)∪(33,+∞)
{x=0x∈(−∞,−33)∪(33,+∞){x2<01−3x2>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=0x∈(−∞,−33)∪(33,+∞){x∈/R1−3x2>0
Solve the inequality
More Steps
Evaluate
1−3x2>0
Rewrite the expression
−3x2>−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2<1
Divide both sides
33x2<31
Divide the numbers
x2<31
Take the 2-th root on both sides of the inequality
x2<31
Calculate
∣x∣<33
Separate the inequality into 2 possible cases
{x<33x>−33
Find the intersection
−33<x<33
{x=0x∈(−∞,−33)∪(33,+∞){x∈/R−33<x<33
Find the intersection
x∈(−∞,−33)∪(33,+∞){x∈/R−33<x<33
Find the intersection
x∈(−∞,−33)∪(33,+∞)x∈/R
Find the union
x∈(−∞,−33)∪(33,+∞)
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)−(x−1−x2)−(x2−3x4)≤0,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
−(x−1−x2)−(x2−3x4)≤0
Remove the parentheses
−x+1+x2−x2+3x4≤0
Simplify the expression
−x+1+3x4≤0
Rewrite the expression
−x+1+3x4=0
Find the critical values by solving the corresponding equation
x∈/R
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x−1−x2<0,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2<0
Move the constant to the right side
x−x2<0−(−1)
Add the terms
x−x2<1
Evaluate
x2−x>−1
Add the same value to both sides
x2−x+41>−1+41
Evaluate
x2−x+41>−43
Evaluate
(x−21)2>−43
Calculate
x∈R
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,x2−3x4≥0−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x2−3x4≥0
Factor the expression
x2(1−3x2)≥0
Separate the inequality into 2 possible cases
{x2≥01−3x2≥0{x2≤01−3x2≤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
{x∈R1−3x2≥0{x2≤01−3x2≤0
Solve the inequality
More Steps
Evaluate
1−3x2≥0
Rewrite the expression
−3x2≥−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2≤1
Divide both sides
33x2≤31
Divide the numbers
x2≤31
Take the 2-th root on both sides of the inequality
x2≤31
Calculate
∣x∣≤33
Separate the inequality into 2 possible cases
{x≤33x≥−33
Find the intersection
−33≤x≤33
{x∈R−33≤x≤33{x2≤01−3x2≤0
Solve the inequality
More Steps
Evaluate
x2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true only when x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
{x∈R−33≤x≤33{x=01−3x2≤0
Solve the inequality
More Steps
Evaluate
1−3x2≤0
Rewrite the expression
−3x2≤−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2≥1
Divide both sides
33x2≥31
Divide the numbers
x2≥31
Take the 2-th root on both sides of the inequality
x2≥31
Calculate
∣x∣≥33
Separate the inequality into 2 possible cases
x≥33x≤−33
Find the union
x∈(−∞,−33]∪[33,+∞)
{x∈R−33≤x≤33{x=0x∈(−∞,−33]∪[33,+∞)
Find the intersection
−33≤x≤33{x=0x∈(−∞,−33]∪[33,+∞)
Find the intersection
−33≤x≤33x∈∅
Find the union
−33≤x≤33
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,−33≤x≤33−(x−1−x2)−(−(x2−3x4))≤0,x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
−(x−1−x2)−(−(x2−3x4))≤0
Remove the parentheses
−x+1+x2+x2−3x4≤0
Simplify the expression
−x+1+2x2−3x4≤0
Rewrite the expression
−x+1+2x2−3x4=0
Find the critical values by solving the corresponding equation
x≈−1.111816x≈0.855758
Determine the test intervals using the critical values
x<−1.111816−1.111816<x<0.855758x>0.855758
Choose a value form each interval
x1=−2x2=0x3=2
To determine if x<−1.111816 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
More Steps
Evaluate
−(−2)+1+2(−2)2−3(−2)4≤0
Simplify
−37≤0
Check the inequality
true
x<−1.111816 is the solutionx2=0x3=2
To determine if −1.111816<x<0.855758 is the solution to the inequality,test if the chosen value x=0 satisfies the initial inequality
More Steps
Evaluate
0+1+2×02−3×04≤0
Simplify
1≤0
Check the inequality
false
x<−1.111816 is the solution−1.111816<x<0.855758 is not a solutionx3=2
To determine if x>0.855758 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
−2+1+2×22−3×24≤0
Simplify
−41≤0
Check the inequality
true
x<−1.111816 is the solution−1.111816<x<0.855758 is not a solutionx>0.855758 is the solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
x≤−1.111816 is the solutionx≥0.855758 is the solution
The final solution of the original inequality is x∈(−∞,−1.111816]∪[0.855758,+∞)
x∈(−∞,−1.111816]∪[0.855758,+∞)
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,−33≤x≤33x∈(−∞,−1.111816]∪[0.855758,+∞),x−1−x2<0,x2−3x4<0
Evaluate
More Steps
Evaluate
x−1−x2<0
Move the constant to the right side
x−x2<0−(−1)
Add the terms
x−x2<1
Evaluate
x2−x>−1
Add the same value to both sides
x2−x+41>−1+41
Evaluate
x2−x+41>−43
Evaluate
(x−21)2>−43
Calculate
x∈R
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,−33≤x≤33x∈(−∞,−1.111816]∪[0.855758,+∞),x∈R,x2−3x4<0
Evaluate
More Steps
Evaluate
x2−3x4<0
Factor the expression
x2(1−3x2)<0
Separate the inequality into 2 possible cases
{x2>01−3x2<0{x2<01−3x2>0
Solve the inequality
More Steps
Evaluate
x2>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 x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
Exclude the impossible values of x
x=0
{x=01−3x2<0{x2<01−3x2>0
Solve the inequality
More Steps
Evaluate
1−3x2<0
Rewrite the expression
−3x2<−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2>1
Divide both sides
33x2>31
Divide the numbers
x2>31
Take the 2-th root on both sides of the inequality
x2>31
Calculate
∣x∣>33
Separate the inequality into 2 possible cases
x>33x<−33
Find the union
x∈(−∞,−33)∪(33,+∞)
{x=0x∈(−∞,−33)∪(33,+∞){x2<01−3x2>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=0x∈(−∞,−33)∪(33,+∞){x∈/R1−3x2>0
Solve the inequality
More Steps
Evaluate
1−3x2>0
Rewrite the expression
−3x2>−1
Change the signs on both sides of the inequality and flip the inequality sign
3x2<1
Divide both sides
33x2<31
Divide the numbers
x2<31
Take the 2-th root on both sides of the inequality
x2<31
Calculate
∣x∣<33
Separate the inequality into 2 possible cases
{x<33x>−33
Find the intersection
−33<x<33
{x=0x∈(−∞,−33)∪(33,+∞){x∈/R−33<x<33
Find the intersection
x∈(−∞,−33)∪(33,+∞){x∈/R−33<x<33
Find the intersection
x∈(−∞,−33)∪(33,+∞)x∈/R
Find the union
x∈(−∞,−33)∪(33,+∞)
−1.111816≤x≤0.855758,x∈/R,−33≤x≤33x∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,−33≤x≤33x∈(−∞,−1.111816]∪[0.855758,+∞),x∈R,x∈(−∞,−33)∪(33,+∞)
Find the intersection
x∈/Rx∈R,x∈/R,x∈(−∞,−33)∪(33,+∞)x∈/R,x∈R,−33≤x≤33x∈(−∞,−1.111816]∪[0.855758,+∞),x∈R,x∈(−∞,−33)∪(33,+∞)
Find the intersection
x∈/Rx∈/Rx∈/R,x∈R,−33≤x≤33x∈(−∞,−1.111816]∪[0.855758,+∞),x∈R,x∈(−∞,−33)∪(33,+∞)
Find the intersection
x∈/Rx∈/Rx∈/Rx∈(−∞,−1.111816]∪[0.855758,+∞),x∈R,x∈(−∞,−33)∪(33,+∞)
Find the intersection
x∈/Rx∈/Rx∈/Rx∈(−∞,−1.111816]∪[0.855758,+∞)
Solution
x∈(−∞,−1.111816]∪[0.855758,+∞)
Show Solution
