Algebra
Calculus
Trigonometry
Matrix
Question :
fracx^{4 + x^2 + 1}x^{2 - 4 x - 5} < 0
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
−1<x<5
Alternative Form
x∈(−1,5)
Evaluate
x2−4x−5x4+x2+1<0
Find the domain
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Evaluate
x2−4x−5=0
Move the constant to the right side
x2−4x=0−(−5)
Add the terms
x2−4x=5
Add the same value to both sides
x2−4x+4=5+4
Evaluate
x2−4x+4=9
Evaluate
(x−2)2=9
Take the root of both sides of the equation and remember to use both positive and negative roots
x−2=±9
Simplify the expression
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Evaluate
9
Write the number in exponential form with the base of 3
32
Reduce the index of the radical and exponent with 2
3
x−2=±3
Separate the inequality into 2 possible cases
{x−2=3x−2=−3
Calculate
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Evaluate
x−2=3
Move the constant to the right side
x=3+2
Add the numbers
x=5
{x=5x−2=−3
Calculate
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Evaluate
x−2=−3
Move the constant to the right side
x=−3+2
Add the numbers
x=−1
{x=5x=−1
Find the intersection
x∈(−∞,−1)∪(−1,5)∪(5,+∞)
x2−4x−5x4+x2+1<0,x∈(−∞,−1)∪(−1,5)∪(5,+∞)
Set the numerator and denominator of x2−4x−5x4+x2+1 equal to 0 to find the values of x where sign changes may occur
x4+x2+1=0x2−4x−5=0
Since the left-hand side is always positive,and the right-hand side is always 0,the statement is false for any value of x
x∈/Rx2−4x−5=0
Calculate
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Evaluate
x2−4x−5=0
Factor the expression
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Evaluate
x2−4x−5
Rewrite the expression
x2+(1−5)x−5
Calculate
x2+x−5x−5
Rewrite the expression
x×x+x−5x−5
Factor out x from the expression
x(x+1)−5x−5
Factor out −5 from the expression
x(x+1)−5(x+1)
Factor out x+1 from the expression
(x−5)(x+1)
(x−5)(x+1)=0
When the product of factors equals 0,at least one factor is 0
x−5=0x+1=0
Solve the equation for x
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Evaluate
x−5=0
Move the constant to the right-hand side and change its sign
x=0+5
Removing 0 doesn't change the value,so remove it from the expression
x=5
x=5x+1=0
Solve the equation for x
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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=5x=−1
x∈/Rx=5x=−1
Determine the test intervals using the critical values
x<−1−1<x<5x>5
Choose a value form each interval
x1=−2x2=2x3=6
To determine if x<−1 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
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Evaluate
(−2)2−4(−2)−5(−2)4+(−2)2+1<0
Simplify
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Evaluate
(−2)2−4(−2)−5(−2)4+(−2)2+1
Multiply the numbers
(−2)2+8−5(−2)4+(−2)2+1
Add the numbers
(−2)2+8−521
Calculate the sum or difference
721
Reduce the numbers
13
Calculate
3
3<0
Check the inequality
false
x<−1 is not a solutionx2=2x3=6
To determine if −1<x<5 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
22−4×2−524+22+1<0
Simplify
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Evaluate
22−4×2−524+22+1
Multiply the numbers
22−8−524+22+1
Add the numbers
22−8−521
Subtract the numbers
−921
Cancel out the common factor 3
−37
Use b−a=−ba=−ba to rewrite the fraction
−37
−37<0
Calculate
−2.3˙<0
Check the inequality
true
x<−1 is not a solution−1<x<5 is the solutionx3=6
To determine if x>5 is the solution to the inequality,test if the chosen value x=6 satisfies the initial inequality
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Evaluate
62−4×6−564+62+1<0
Simplify
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Evaluate
62−4×6−564+62+1
Multiply the numbers
62−24−564+62+1
Add the numbers
62−24−51333
Subtract the numbers
71333
71333<0
Calculate
190.4˙28571˙<0
Check the inequality
false
x<−1 is not a solution−1<x<5 is the solutionx>5 is not a solution
The original inequality is a strict inequality,so does not include the critical value ,the final solution is −1<x<5
−1<x<5
Check if the solution is in the defined range
−1<x<5,x∈(−∞,−1)∪(−1,5)∪(5,+∞)
Solution
−1<x<5
Alternative Form
x∈(−1,5)
Show Solution
