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∈(1,2)∪(2,3)
Evaluate
(x−1)(3−x)(x−2)2>0
Rewrite the expression
(x−1)(3−x)(x−2)2=0
Separate the equation into 3 possible cases
x−1=03−x=0(x−2)2=0
Solve the equation
More Steps
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=13−x=0(x−2)2=0
Solve the equation
More Steps
Evaluate
3−x=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
Change the signs on both sides of the equation
x=3
x=1x=3(x−2)2=0
Solve the equation
More Steps
Evaluate
(x−2)2=0
The only way a power can be 0 is when the base equals 0
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=1x=3x=2
Determine the test intervals using the critical values
x<11<x<22<x<3x>3
Choose a value form each interval
x1=0x2=23x3=25x4=4
To determine if x<1 is the solution to the inequality,test if the chosen value x=0 satisfies the initial inequality
More Steps
Evaluate
(0−1)(3−0)(0−2)2>0
Simplify
More Steps
Evaluate
(0−1)(3−0)(0−2)2
Removing 0 doesn't change the value,so remove it from the expression
(−1)(3−0)(0−2)2
Remove the parentheses
−(3−0)(0−2)2
Removing 0 doesn't change the value,so remove it from the expression
−3(0−2)2
Removing 0 doesn't change the value,so remove it from the expression
−3(−2)2
Multiply the terms
−12
−12>0
Check the inequality
false
x<1 is not a solutionx2=23x3=25x4=4
To determine if 1<x<2 is the solution to the inequality,test if the chosen value x=23 satisfies the initial inequality
More Steps
Evaluate
(23−1)(3−23)(23−2)2>0
Simplify
More Steps
Evaluate
(23−1)(3−23)(23−2)2
Subtract the numbers
21(3−23)(23−2)2
Subtract the numbers
21×23(23−2)2
Subtract the numbers
21×23(−21)2
Multiply the terms
43(−21)2
Evaluate the power
43×221
To multiply the fractions,multiply the numerators and denominators separately
4×223
Multiply the numbers
243
243>0
Calculate
0.1875>0
Check the inequality
true
x<1 is not a solution1<x<2 is the solutionx3=25x4=4
To determine if 2<x<3 is the solution to the inequality,test if the chosen value x=25 satisfies the initial inequality
More Steps
Evaluate
(25−1)(3−25)(25−2)2>0
Simplify
More Steps
Evaluate
(25−1)(3−25)(25−2)2
Subtract the numbers
23(3−25)(25−2)2
Subtract the numbers
23×21(25−2)2
Subtract the numbers
23×21(21)2
Multiply the terms with the same base by adding their exponents
23(21)1+2
Add the numbers
23(21)3
Evaluate the power
23×231
To multiply the fractions,multiply the numerators and denominators separately
2×233
Multiply the numbers
243
243>0
Calculate
0.1875>0
Check the inequality
true
x<1 is not a solution1<x<2 is the solution2<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
(4−1)(3−4)(4−2)2>0
Simplify
More Steps
Evaluate
(4−1)(3−4)(4−2)2
Subtract the numbers
3(3−4)(4−2)2
Subtract the numbers
3(−1)(4−2)2
Subtract the numbers
3(−1)×22
Any expression multiplied by 1 remains the same
−3×22
Multiply the terms
−12
−12>0
Check the inequality
false
x<1 is not a solution1<x<2 is the solution2<x<3 is the solutionx>3 is not a solution
Solution
x∈(1,2)∪(2,3)
Show Solution
