Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,0)∪(0,1]∪(3,+∞)∪{2}
Evaluate
x2(x−3)(x−1)1(x−2)2(x4)1≥0
Find the domain
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Evaluate
x2(x−3)=0
Apply the zero product property
{x2=0x−3=0
The only way a power can not be 0 is when the base not equals 0
{x=0x−3=0
Solve the inequality
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Evaluate
x−3=0
Move the constant to the right side
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
{x=0x=3
Find the intersection
x∈(−∞,0)∪(0,3)∪(3,+∞)
x2(x−3)(x−1)1(x−2)2(x4)1≥0,x∈(−∞,0)∪(0,3)∪(3,+∞)
Simplify
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Evaluate
x2(x−3)(x−1)1(x−2)2(x4)1
Multiply the exponents
x2(x−3)(x−1)1(x−2)2x4×1
Calculate
x2(x−3)(x−1)(x−2)2x4×1
Any expression multiplied by 1 remains the same
x2(x−3)(x−1)(x−2)2x4
Reduce the fraction
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Calculate
x2x4
Use the product rule aman=an−m to simplify the expression
x4−2
Subtract the terms
x2
x−3(x−1)(x−2)2x2
Use the commutative property to reorder the terms
x−3(x−1)x2(x−2)2
Multiply the first two terms
x−3x2(x−1)(x−2)2
x−3x2(x−1)(x−2)2≥0
Separate the inequality into 2 possible cases
{x2(x−1)(x−2)2≥0x−3>0{x2(x−1)(x−2)2≤0x−3<0
Solve the inequality
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Evaluate
x2(x−1)(x−2)2≥0
Separate the inequality into 2 possible cases
{x2≥0(x−1)(x−2)2≥0{x2≤0(x−1)(x−2)2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x∈R(x−1)(x−2)2≥0{x2≤0(x−1)(x−2)2≤0
Solve the inequality
More Steps
Evaluate
(x−1)(x−2)2≥0
Separate the inequality into 2 possible cases
{x−1≥0(x−2)2≥0{x−1≤0(x−2)2≤0
Solve the inequality
{x≥1(x−2)2≥0{x−1≤0(x−2)2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x≥1x∈R{x−1≤0(x−2)2≤0
Solve the inequality
{x≥1x∈R{x≤1(x−2)2≤0
Solve the inequality
{x≥1x∈R{x≤1x=2
Find the intersection
x≥1{x≤1x=2
Find the intersection
x≥1x∈∅
Find the union
x≥1
{x∈Rx≥1{x2≤0(x−1)(x−2)2≤0
Solve the inequality
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Evaluate
x2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true only when x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
{x∈Rx≥1{x=0(x−1)(x−2)2≤0
Solve the inequality
More Steps
Evaluate
(x−1)(x−2)2≤0
Separate the inequality into 2 possible cases
{x−1≥0(x−2)2≤0{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1(x−2)2≤0{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1x=2{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1x=2{x≤1(x−2)2≥0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x≥1x=2{x≤1x∈R
Find the intersection
x=2{x≤1x∈R
Find the intersection
x=2x≤1
Find the union
x∈(−∞,1]∪{2}
{x∈Rx≥1{x=0x∈(−∞,1]∪{2}
Find the intersection
x≥1{x=0x∈(−∞,1]∪{2}
Find the intersection
x≥1x=0
Find the union
x∈[1,+∞)∪{0}
{x∈[1,+∞)∪{0}x−3>0{x2(x−1)(x−2)2≤0x−3<0
Solve the inequality
More Steps
Evaluate
x−3>0
Move the constant to the right side
x>0+3
Removing 0 doesn't change the value,so remove it from the expression
x>3
{x∈[1,+∞)∪{0}x>3{x2(x−1)(x−2)2≤0x−3<0
Solve the inequality
More Steps
Evaluate
x2(x−1)(x−2)2≤0
Separate the inequality into 2 possible cases
{x2≥0(x−1)(x−2)2≤0{x2≤0(x−1)(x−2)2≥0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x∈R(x−1)(x−2)2≤0{x2≤0(x−1)(x−2)2≥0
Solve the inequality
More Steps
Evaluate
(x−1)(x−2)2≤0
Separate the inequality into 2 possible cases
{x−1≥0(x−2)2≤0{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1(x−2)2≤0{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1x=2{x−1≤0(x−2)2≥0
Solve the inequality
{x≥1x=2{x≤1(x−2)2≥0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x≥1x=2{x≤1x∈R
Find the intersection
x=2{x≤1x∈R
Find the intersection
x=2x≤1
Find the union
x∈(−∞,1]∪{2}
{x∈Rx∈(−∞,1]∪{2}{x2≤0(x−1)(x−2)2≥0
Solve the inequality
More Steps
Evaluate
x2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true only when x2=0
x2=0
The only way a power can be 0 is when the base equals 0
x=0
{x∈Rx∈(−∞,1]∪{2}{x=0(x−1)(x−2)2≥0
Solve the inequality
More Steps
Evaluate
(x−1)(x−2)2≥0
Separate the inequality into 2 possible cases
{x−1≥0(x−2)2≥0{x−1≤0(x−2)2≤0
Solve the inequality
{x≥1(x−2)2≥0{x−1≤0(x−2)2≤0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x
{x≥1x∈R{x−1≤0(x−2)2≤0
Solve the inequality
{x≥1x∈R{x≤1(x−2)2≤0
Solve the inequality
{x≥1x∈R{x≤1x=2
Find the intersection
x≥1{x≤1x=2
Find the intersection
x≥1x∈∅
Find the union
x≥1
{x∈Rx∈(−∞,1]∪{2}{x=0x≥1
Find the intersection
x∈(−∞,1]∪{2}{x=0x≥1
Find the intersection
x∈(−∞,1]∪{2}x∈∅
Find the union
x∈(−∞,1]∪{2}
{x∈[1,+∞)∪{0}x>3{x∈(−∞,1]∪{2}x−3<0
Solve the inequality
More Steps
Evaluate
x−3<0
Move the constant to the right side
x<0+3
Removing 0 doesn't change the value,so remove it from the expression
x<3
{x∈[1,+∞)∪{0}x>3{x∈(−∞,1]∪{2}x<3
Find the intersection
x>3{x∈(−∞,1]∪{2}x<3
Find the intersection
x>3x∈(−∞,1]∪{2}
Find the union
x∈(−∞,1]∪(3,+∞)∪{2}
Check if the solution is in the defined range
x∈(−∞,1]∪(3,+∞)∪{2},x∈(−∞,0)∪(0,3)∪(3,+∞)
Solution
x∈(−∞,0)∪(0,1]∪(3,+∞)∪{2}
Show Solution
