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∈[−22+2,0)∪[22+2,+∞)
Evaluate
x−x4−2≥2
Find the domain
x−x4−2≥2,x=0
Move the expression to the left side
x−x4−2−2≥0
Subtract the terms
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Evaluate
x−x4−2−2
Subtract the numbers
x−x4−4
Reduce fractions to a common denominator
xx×x−x4−x4x
Write all numerators above the common denominator
xx×x−4−4x
Multiply the terms
xx2−4−4x
xx2−4−4x≥0
Set the numerator and denominator of xx2−4−4x equal to 0 to find the values of x where sign changes may occur
x2−4−4x=0x=0
Calculate
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Evaluate
x2−4−4x=0
Add or subtract both sides
x2−4x=4
Add the same value to both sides
x2−4x+4=4+4
Simplify the expression
(x−2)2=8
Take the root of both sides of the equation and remember to use both positive and negative roots
x−2=±8
Simplify the expression
x−2=±22
Separate the equation into 2 possible cases
x−2=22x−2=−22
Move the constant to the right-hand side and change its sign
x=22+2x−2=−22
Move the constant to the right-hand side and change its sign
x=22+2x=−22+2
x=22+2x=−22+2x=0
Determine the test intervals using the critical values
x<−22+2−22+2<x<00<x<22+2x>22+2
Choose a value form each interval
x1=−2x2=−2+1x3=2x4=6
To determine if x<−22+2 is the solution to the inequality,test if the chosen value x=−2 satisfies the initial inequality
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Evaluate
−2−−24−2≥2
Simplify
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Evaluate
−2−−24−2
Multiplying or dividing an even number of negative terms equals a positive
−2+24−2
Divide the terms
−2+2−2
Apply the inverse property of addition
−2
−2≥2
Check the inequality
false
x<−22+2 is not a solutionx2=−2+1x3=2x4=6
To determine if −22+2<x<0 is the solution to the inequality,test if the chosen value x=−2+1 satisfies the initial inequality
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Evaluate
−2+1−−2+14−2≥2
Calculate the sum or difference
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Evaluate
−2+1−−2+14−2
Subtract the numbers
−2−1−−2+14
Calculate
−2−1−(−42−4)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−2−1+42+4
Add the numbers
32−1+4
Add the numbers
32+3
32+3≥2
Calculate
7.242641≥2
Check the inequality
true
x<−22+2 is not a solution−22+2<x<0 is the solutionx3=2x4=6
To determine if 0<x<22+2 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
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Evaluate
2−24−2≥2
Simplify
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Evaluate
2−24−2
Divide the terms
2−2−2
Apply the inverse property of addition
−2
−2≥2
Check the inequality
false
x<−22+2 is not a solution−22+2<x<0 is the solution0<x<22+2 is not a solutionx4=6
To determine if x>22+2 is the solution to the inequality,test if the chosen value x=6 satisfies the initial inequality
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Evaluate
6−64−2≥2
Simplify
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Evaluate
6−64−2
Cancel out the common factor 2
6−32−2
Subtract the numbers
4−32
Reduce fractions to a common denominator
34×3−32
Write all numerators above the common denominator
34×3−2
Multiply the numbers
312−2
Subtract the numbers
310
310≥2
Calculate
3.3˙≥2
Check the inequality
true
x<−22+2 is not a solution−22+2<x<0 is the solution0<x<22+2 is not a solutionx>22+2 is the solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
−22+2≤x<0 is the solutionx≥22+2 is the solution
The final solution of the original inequality is x∈[−22+2,0)∪[22+2,+∞)
x∈[−22+2,0)∪[22+2,+∞)
Check if the solution is in the defined range
x∈[−22+2,0)∪[22+2,+∞),x=0
Solution
x∈[−22+2,0)∪[22+2,+∞)
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