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∈(−5+1,0)∪(0,2)∪(5+1,+∞)
Evaluate
x−x22<2x−43x−1
Find the domain
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Evaluate
x2=0
The only way a power can not be 0 is when the base not equals 0
x=0
x−x22<2x−43x−1,x=0
Subtract the terms
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Evaluate
2x−43x
Collect like terms by calculating the sum or difference of their coefficients
(2−43)x
Subtract the numbers
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Evaluate
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
45x
x−x22<45x−1
Move the expression to the left side
x−x22−(45x−1)<0
Subtract the terms
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Evaluate
x−x22−(45x−1)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x−x22−45x+1
Subtract the terms
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Evaluate
x−45x
Collect like terms by calculating the sum or difference of their coefficients
(1−45)x
Subtract the numbers
−41x
−41x−x22+1
Rewrite the expression
−4x−x22+1
Reduce fractions to a common denominator
−4x2x×x2−x2×42×4+4x24x2
Use the commutative property to reorder the terms
−4x2x×x2−4x22×4+4x24x2
Write all numerators above the common denominator
4x2−x×x2−2×4+4x2
Multiply the terms
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Evaluate
x×x2
Use the product rule an×am=an+m to simplify the expression
x1+2
Add the numbers
x3
4x2−x3−2×4+4x2
Multiply the numbers
4x2−x3−8+4x2
4x2−x3−8+4x2<0
Change the signs on both sides of the inequality and flip the inequality sign
4x2x3+8−4x2>0
Set the numerator and denominator of 4x2x3+8−4x2 equal to 0 to find the values of x where sign changes may occur
x3+8−4x2=04x2=0
Calculate
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Evaluate
x3+8−4x2=0
Factor the expression
(x−2)(x2−4−2x)=0
Separate the equation into 2 possible cases
x−2=0x2−4−2x=0
Solve the equation
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Evaluate
x−2=0
Move the constant to the right-hand side and change its sign
x=0+2
Removing 0 doesn't change the value,so remove it from the expression
x=2
x=2x2−4−2x=0
Solve the equation
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Evaluate
x2−4−2x=0
Add or subtract both sides
x2−2x=4
Add the same value to both sides
x2−2x+1=4+1
Simplify the expression
(x−1)2=5
Take the root of both sides of the equation and remember to use both positive and negative roots
x−1=±5
Separate the equation into 2 possible cases
x−1=5x−1=−5
Move the constant to the right-hand side and change its sign
x=5+1x−1=−5
Move the constant to the right-hand side and change its sign
x=5+1x=−5+1
x=2x=5+1x=−5+1
x=2x=5+1x=−5+14x2=0
Calculate
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Evaluate
4x2=0
Rewrite the expression
x2=0
The only way a power can be 0 is when the base equals 0
x=0
x=2x=5+1x=−5+1x=0
Determine the test intervals using the critical values
x<−5+1−5+1<x<00<x<22<x<5+1x>5+1
Choose a value form each interval
x1=−2x2=−1x3=1x4=3x5=4
To determine if x<−5+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)22<45(−2)−1
Simplify
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Evaluate
−2−(−2)22
Divide the terms
−2−21
Reduce fractions to a common denominator
−22×2−21
Write all numerators above the common denominator
2−2×2−1
Multiply the numbers
2−4−1
Subtract the numbers
2−5
Use b−a=−ba=−ba to rewrite the fraction
−25
−25<45(−2)−1
Simplify
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Evaluate
45(−2)−1
Multiply the numbers
−25−1
Reduce fractions to a common denominator
−25−22
Write all numerators above the common denominator
2−5−2
Subtract the numbers
2−7
Use b−a=−ba=−ba to rewrite the fraction
−27
−25<−27
Calculate
−2.5<−27
Calculate
−2.5<−3.5
Check the inequality
false
x<−5+1 is not a solutionx2=−1x3=1x4=3x5=4
To determine if −5+1<x<0 is the solution to the inequality,test if the chosen value x=−1 satisfies the initial inequality
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Evaluate
−1−(−1)22<45(−1)−1
Simplify
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Evaluate
−1−(−1)22
Evaluate the power
−1−12
Divide the terms
−1−2
Subtract the numbers
−3
−3<45(−1)−1
Simplify
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Evaluate
45(−1)−1
Multiplying or dividing an odd number of negative terms equals a negative
−45−1
Reduce fractions to a common denominator
−45−44
Write all numerators above the common denominator
4−5−4
Subtract the numbers
4−9
Use b−a=−ba=−ba to rewrite the fraction
−49
−3<−49
Calculate
−3<−2.25
Check the inequality
true
x<−5+1 is not a solution−5+1<x<0 is the solutionx3=1x4=3x5=4
To determine if 0<x<2 is the solution to the inequality,test if the chosen value x=1 satisfies the initial inequality
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Evaluate
1−122<45×1−1
Simplify
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Evaluate
1−122
1 raised to any power equals to 1
1−12
Divide the terms
1−2
Subtract the numbers
−1
−1<45×1−1
Simplify
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Evaluate
45×1−1
Any expression multiplied by 1 remains the same
45−1
Reduce fractions to a common denominator
45−44
Write all numerators above the common denominator
45−4
Subtract the numbers
41
−1<41
Calculate
−1<0.25
Check the inequality
true
x<−5+1 is not a solution−5+1<x<0 is the solution0<x<2 is the solutionx4=3x5=4
To determine if 2<x<5+1 is the solution to the inequality,test if the chosen value x=3 satisfies the initial inequality
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Evaluate
3−322<45×3−1
Subtract the numbers
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Evaluate
3−322
Evaluate the power
3−92
Reduce fractions to a common denominator
93×9−92
Write all numerators above the common denominator
93×9−2
Multiply the numbers
927−2
Subtract the numbers
925
925<45×3−1
Simplify
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Evaluate
45×3−1
Multiply the numbers
415−1
Reduce fractions to a common denominator
415−44
Write all numerators above the common denominator
415−4
Subtract the numbers
411
925<411
Calculate
2.7˙<411
Calculate
2.7˙<2.75
Check the inequality
false
x<−5+1 is not a solution−5+1<x<0 is the solution0<x<2 is the solution2<x<5+1 is not a solutionx5=4
To determine if x>5+1 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
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Evaluate
4−422<45×4−1
Simplify
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Evaluate
4−422
Divide the terms
4−231
Evaluate the power
4−81
Reduce fractions to a common denominator
84×8−81
Write all numerators above the common denominator
84×8−1
Multiply the numbers
832−1
Subtract the numbers
831
831<45×4−1
Simplify
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Evaluate
45×4−1
Multiply the numbers
5−1
Subtract the numbers
4
831<4
Calculate
3.875<4
Check the inequality
true
x<−5+1 is not a solution−5+1<x<0 is the solution0<x<2 is the solution2<x<5+1 is not a solutionx>5+1 is the solution
The original inequality is a strict inequality,so does not include the critical value ,the final solution is x∈(−5+1,0)∪(0,2)∪(5+1,+∞)
x∈(−5+1,0)∪(0,2)∪(5+1,+∞)
Check if the solution is in the defined range
x∈(−5+1,0)∪(0,2)∪(5+1,+∞),x=0
Solution
x∈(−5+1,0)∪(0,2)∪(5+1,+∞)
Show Solution
