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)∪(23,3)
Evaluate
x×x2×2(x−1)×5(2x−3)(x−3)<0
Multiply
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Evaluate
x×x2×2(x−1)×5(2x−3)(x−3)
Multiply the terms with the same base by adding their exponents
x1+2×2(x−1)×5(2x−3)(x−3)
Add the numbers
x3×2(x−1)×5(2x−3)(x−3)
Multiply the terms
x3×10(x−1)(2x−3)(x−3)
Use the commutative property to reorder the terms
10x3(x−1)(2x−3)(x−3)
10x3(x−1)(2x−3)(x−3)<0
Rewrite the expression
10x3(x−1)(2x−3)(x−3)=0
Elimination the left coefficient
x3(x−1)(2x−3)(x−3)=0
Separate the equation into 4 possible cases
x3=0x−1=02x−3=0x−3=0
The only way a power can be 0 is when the base equals 0
x=0x−1=02x−3=0x−3=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=12x−3=0x−3=0
Solve the equation
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Evaluate
2x−3=0
Move the constant to the right-hand side and change its sign
2x=0+3
Removing 0 doesn't change the value,so remove it from the expression
2x=3
Divide both sides
22x=23
Divide the numbers
x=23
x=0x=1x=23x−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=23x=3
Determine the test intervals using the critical values
x<00<x<11<x<2323<x<3x>3
Choose a value form each interval
x1=−1x2=21x3=45x4=2x5=4
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
10(−1)3(−1−1)(2(−1)−3)(−1−3)<0
Simplify
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Evaluate
10(−1)3(−1−1)(2(−1)−3)(−1−3)
Subtract the numbers
10(−1)3(−2)(2(−1)−3)(−1−3)
Simplify
10(−1)3(−2)(−2−3)(−1−3)
Subtract the numbers
10(−1)3(−2)(−5)(−1−3)
Subtract the numbers
10(−1)3(−2)(−5)(−4)
Rewrite the expression
−10(−1)3×2×5×4
Multiply the terms
−10(−1)3×40
Multiply the terms
−(−400)
When there is - in front of an expression in parentheses change the sign of each term of the expression and remove the parentheses
400
400<0
Check the inequality
false
x<0 is not a solutionx2=21x3=45x4=2x5=4
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
10(21)3(21−1)(2×21−3)(21−3)<0
Simplify
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Evaluate
10(21)3(21−1)(2×21−3)(21−3)
Subtract the numbers
10(21)3(−21)(2×21−3)(21−3)
Multiply the numbers
10(21)3(−21)(1−3)(21−3)
Subtract the numbers
10(21)3(−21)(−2)(21−3)
Subtract the numbers
10(21)3(−21)(−2)(−25)
Rewrite the expression
−10(21)3×21×2×25
Transform the expression
−10(21)3×21(21)−1×25
Multiply the terms with the same base by adding their exponents
−10(21)3+1−1×25
Calculate the sum or difference
−10(21)3×25
Multiply the terms
−825
−825<0
Calculate
−3.125<0
Check the inequality
true
x<0 is not a solution0<x<1 is the solutionx3=45x4=2x5=4
To determine if 1<x<23 is the solution to the inequality,test if the chosen value x=45 satisfies the initial inequality
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Evaluate
10(45)3(45−1)(2×45−3)(45−3)<0
Simplify
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Evaluate
10(45)3(45−1)(2×45−3)(45−3)
Subtract the numbers
10(45)3×41(2×45−3)(45−3)
Multiply the numbers
10(45)3×41(25−3)(45−3)
Subtract the numbers
10(45)3×41(−21)(45−3)
Subtract the numbers
10(45)3×41(−21)(−47)
Rewrite the expression
10(45)3×41×21×47
Multiply the terms
10(45)3×327
Multiply the terms
32625×327
To multiply the fractions,multiply the numerators and denominators separately
32×32625×7
Multiply the numbers
32×324375
Multiply the numbers
10244375
10244375<0
Calculate
4.272461<0
Check the inequality
false
x<0 is not a solution0<x<1 is the solution1<x<23 is not a solutionx4=2x5=4
To determine if 23<x<3 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
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Evaluate
10×23(2−1)(2×2−3)(2−3)<0
Simplify
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Evaluate
10×23(2−1)(2×2−3)(2−3)
Subtract the numbers
10×23×1×(2×2−3)(2−3)
Multiply the numbers
10×23×1×(4−3)(2−3)
Subtract the numbers
10×23×1×1×(2−3)
Subtract the numbers
10×23×1×1×(−1)
Rewrite the expression
10×23(−1)
Any expression multiplied by 1 remains the same
−10×23
Multiply the terms
−80
−80<0
Check the inequality
true
x<0 is not a solution0<x<1 is the solution1<x<23 is not a solution23<x<3 is the solutionx5=4
To determine if x>3 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
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Evaluate
10×43(4−1)(2×4−3)(4−3)<0
Simplify
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Evaluate
10×43(4−1)(2×4−3)(4−3)
Subtract the numbers
10×43×3(2×4−3)(4−3)
Multiply the numbers
10×43×3(8−3)(4−3)
Subtract the numbers
10×43×3×5(4−3)
Subtract the numbers
10×43×3×5×1
Rewrite the expression
10×43×3×5
Multiply the terms
10×43×15
Multiply the terms
640×15
Multiply the numbers
9600
9600<0
Check the inequality
false
x<0 is not a solution0<x<1 is the solution1<x<23 is not a solution23<x<3 is the solutionx>3 is not a solution
Solution
x∈(0,1)∪(23,3)
Show Solution
