Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
x∈(−1,0)∪(3,+∞)
Evaluate
x−x3−2>0
Find the domain
x−x3−2>0,x=0
Rearrange the terms
xx2−3−2x>0
Set the numerator and denominator of xx2−3−2x equal to 0 to find the values of x where sign changes may occur
x2−3−2x=0x=0
Calculate
More Steps
Evaluate
x2−3−2x=0
Factor the expression
More Steps
Evaluate
x2−3−2x
Reorder the terms
x2−2x−3
Rewrite the expression
x2+(1−3)x−3
Calculate
x2+x−3x−3
Rewrite the expression
x×x+x−3x−3
Factor out x from the expression
x(x+1)−3x−3
Factor out −3 from the expression
x(x+1)−3(x+1)
Factor out x+1 from the expression
(x−3)(x+1)
(x−3)(x+1)=0
When the product of factors equals 0,at least one factor is 0
x−3=0x+1=0
Solve the equation for x
More Steps
Evaluate
x−3=0
Move the constant to the right-hand side and change its sign
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x=3x+1=0
Solve the equation for x
More Steps
Evaluate
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
x=3x=−1
x=3x=−1x=0
Determine the test intervals using the critical values
x<−1−1<x<00<x<3x>3
Choose a value form each interval
x1=−2x2=−21x3=2x4=4
To determine if x<−1 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
More Steps
Evaluate
−2−−23−2>0
Simplify
More Steps
Evaluate
−2−−23−2
Multiplying or dividing an even number of negative terms equals a positive
−2+23−2
Subtract the numbers
−4+23
Reduce fractions to a common denominator
−24×2+23
Write all numerators above the common denominator
2−4×2+3
Multiply the numbers
2−8+3
Add the numbers
2−5
Use b−a=−ba=−ba to rewrite the fraction
−25
−25>0
Calculate
−2.5>0
Check the inequality
false
x<−1 is not a solutionx2=−21x3=2x4=4
To determine if −1<x<0 is the solution to the inequality,test if the chosen value x=−21 satisfies the initial inequality
More Steps
Evaluate
−21−−213−2>0
Simplify
More Steps
Evaluate
−21−−213−2
Multiplying or dividing an even number of negative terms equals a positive
−21+213−2
Divide the terms
−21+6−2
Subtract the numbers
−21+4
Reduce fractions to a common denominator
−21+24×2
Write all numerators above the common denominator
2−1+4×2
Multiply the numbers
2−1+8
Add the numbers
27
27>0
Calculate
3.5>0
Check the inequality
true
x<−1 is not a solution−1<x<0 is the solutionx3=2x4=4
To determine if 0<x<3 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
2−23−2>0
Since two opposites add up to 0,remove them form the expression
−23>0
Calculate
−1.5>0
Check the inequality
false
x<−1 is not a solution−1<x<0 is the solution0<x<3 is not a solutionx4=4
To determine if x>3 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
More Steps
Evaluate
4−43−2>0
Subtract the numbers
More Steps
Evaluate
4−43−2
Subtract the numbers
2−43
Reduce fractions to a common denominator
42×4−43
Write all numerators above the common denominator
42×4−3
Multiply the numbers
48−3
Subtract the numbers
45
45>0
Calculate
1.25>0
Check the inequality
true
x<−1 is not a solution−1<x<0 is the solution0<x<3 is not a solutionx>3 is the solution
The original inequality is a strict inequality,so does not include the critical value ,the final solution is x∈(−1,0)∪(3,+∞)
x∈(−1,0)∪(3,+∞)
Check if the solution is in the defined range
x∈(−1,0)∪(3,+∞),x=0
Solution
x∈(−1,0)∪(3,+∞)
Show Solution
