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)∪(1,3)
Evaluate
x5(x−1)2(x−3)<0
Rewrite the expression
x5(x−1)2(x−3)=0
Separate the equation into 3 possible cases
x5=0(x−1)2=0x−3=0
The only way a power can be 0 is when the base equals 0
x=0(x−1)2=0x−3=0
Solve the equation
More Steps
Evaluate
(x−1)2=0
The only way a power can be 0 is when the base equals 0
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=1x−3=0
Solve the equation
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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=0x=1x=3
Determine the test intervals using the critical values
x<00<x<11<x<3x>3
Choose a value form each interval
x1=−1x2=21x3=2x4=4
To determine if x<0 is the solution to the inequality,test if the chosen value x=−1 satisfies the initial inequality
More Steps
Evaluate
(−1)5(−1−1)2(−1−3)<0
Simplify
More Steps
Evaluate
(−1)5(−1−1)2(−1−3)
Subtract the numbers
(−1)5(−1−1)2(−4)
Subtract the numbers
(−1)5(−2)2(−4)
Rewrite the expression
−(−1)5(−2)2×4
Multiply the terms
−(−16)
When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses
16
16<0
Check the inequality
false
x<0 is not a solutionx2=21x3=2x4=4
To determine if 0<x<1 is the solution to the inequality,test if the chosen value x=21 satisfies the initial inequality
More Steps
Evaluate
(21)5(21−1)2(21−3)<0
Simplify
More Steps
Evaluate
(21)5(21−1)2(21−3)
Subtract the numbers
(21)5(21−1)2(−25)
Subtract the numbers
(21)5(−21)2(−25)
Rewrite the expression
−(21)5(−21)2×25
Multiply the terms
−285
−285<0
Calculate
−0.01953125<0
Check the inequality
true
x<0 is not a solution0<x<1 is the solutionx3=2x4=4
To determine if 1<x<3 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
25(2−1)2(2−3)<0
Simplify
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Evaluate
25(2−1)2(2−3)
Subtract the numbers
25(2−1)2(−1)
Subtract the numbers
25×12×(−1)
1 raised to any power equals to 1
25×1×(−1)
Rewrite the expression
25(−1)
Multiply the terms
−25
−25<0
Calculate
−32<0
Check the inequality
true
x<0 is not a solution0<x<1 is the solution1<x<3 is the 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
45(4−1)2(4−3)<0
Simplify
More Steps
Evaluate
45(4−1)2(4−3)
Subtract the numbers
45(4−1)2×1
Subtract the numbers
45×32×1
Rewrite the expression
45×32
Expand the expression
1024×32
Expand the expression
1024×9
Multiply the numbers
9216
9216<0
Check the inequality
false
x<0 is not a solution0<x<1 is the solution1<x<3 is the solutionx>3 is not a solution
Solution
x∈(0,1)∪(1,3)
Show Solution
