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∈(−∞,0]∪[1,+∞)
Evaluate
(x−5)2(x×1)3(x−1)≥0
Simplify
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Evaluate
(x−5)2(x×1)3(x−1)
Any expression multiplied by 1 remains the same
(x−5)2x3(x−1)
Use the commutative property to reorder the terms
x3(x−5)2(x−1)
x3(x−5)2(x−1)≥0
Rewrite the expression
x3(x−5)2(x−1)=0
Separate the equation into 3 possible cases
x3=0(x−5)2=0x−1=0
The only way a power can be 0 is when the base equals 0
x=0(x−5)2=0x−1=0
Solve the equation
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Evaluate
(x−5)2=0
The only way a power can be 0 is when the base equals 0
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=0x=5x−1=0
Solve the equation
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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=0x=5x=1
Determine the test intervals using the critical values
x<00<x<11<x<5x>5
Choose a value form each interval
x1=−1x2=21x3=3x4=6
To determine if 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)3(−1−5)2(−1−1)≥0
Simplify
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Evaluate
(−1)3(−1−5)2(−1−1)
Subtract the numbers
(−1)3(−1−5)2(−2)
Subtract the numbers
(−1)3(−6)2(−2)
Rewrite the expression
−(−1)3(−6)2×2
Multiply the terms
−(−72)
When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses
72
72≥0
Check the inequality
true
x<0 is the solutionx2=21x3=3x4=6
To determine if 0<x<1 is the solution to the inequality,test if the chosen value x=21 satisfies the initial inequality
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Evaluate
(21)3(21−5)2(21−1)≥0
Simplify
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Evaluate
(21)3(21−5)2(21−1)
Subtract the numbers
(21)3(21−5)2(−21)
Subtract the numbers
(21)3(−29)2(−21)
Rewrite the expression
−(21)3(−29)2×21
Multiply the terms with the same base by adding their exponents
−(21)3+1(−29)2
Add the numbers
−(21)4(−29)2
Multiply the numbers
−6481
−6481≥0
Calculate
−1.265625≥0
Check the inequality
false
x<0 is the solution0<x<1 is not a solutionx3=3x4=6
To determine if 1<x<5 is the solution to the inequality,test if the chosen value x=3 satisfies the initial inequality
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Evaluate
33(3−5)2(3−1)≥0
Simplify
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Evaluate
33(3−5)2(3−1)
Subtract the numbers
33(3−5)2×2
Subtract the numbers
33(−2)2×2
Multiply the numbers
108×2
Multiply the numbers
216
216≥0
Check the inequality
true
x<0 is the solution0<x<1 is not a solution1<x<5 is the solutionx4=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
63(6−5)2(6−1)≥0
Simplify
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Evaluate
63(6−5)2(6−1)
Subtract the numbers
63(6−5)2×5
Subtract the numbers
63×12×5
1 raised to any power equals to 1
63×1×5
Rewrite the expression
63×5
Evaluate the power
216×5
Multiply the numbers
1080
1080≥0
Check the inequality
true
x<0 is the solution0<x<1 is not a solution1<x<5 is the solutionx>5 is the solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
x≤0 is the solution1≤x≤5 is the solutionx≥5 is the solution
Solution
x∈(−∞,0]∪[1,+∞)
Show Solution
