Question
Solve the inequality
x∈(−∞,0)∪(0,1]∪[2,3]
Evaluate
(x3×x2)(x−3)(x−2)×x−(x−1)≥0
Find the domain
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Evaluate
{x3×x2=0x=0
Calculate
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Evaluate
x3×x2=0
Multiply the terms
x5=0
The only way a power can not be 0 is when the base not equals 0
x=0
{x=0x=0
Find the intersection
x=0
(x3×x2)(x−3)(x−2)×x−(x−1)≥0,x=0
Remove the parentheses
x3×x2(x−3)(x−2)×x−(x−1)≥0
Simplify
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Evaluate
x3×x2(x−3)(x−2)×x−(x−1)
Multiply the terms
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Evaluate
x3×x2
Use the product rule an×am=an+m to simplify the expression
x3+2
Add the numbers
x5
x5(x−3)(x−2)×x−(x−1)
Calculate
x5(x−3)(x−2)×x−x+1
Multiply the terms
x5×x(x−3)(x−2)(−x+1)
Multiply the terms
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Evaluate
x5×x
Use the product rule an×am=an+m to simplify the expression
x5+1
Add the numbers
x6
x6(x−3)(x−2)(−x+1)
x6(x−3)(x−2)(−x+1)≥0
Separate the inequality into 2 possible cases
{(x−3)(x−2)(−x+1)≥0x6>0{(x−3)(x−2)(−x+1)≤0x6<0
Solve the inequality
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Evaluate
(x−3)(x−2)(−x+1)≥0
Separate the inequality into 2 possible cases
{x−3≥0(x−2)(−x+1)≥0{x−3≤0(x−2)(−x+1)≤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≥3(x−2)(−x+1)≥0{x−3≤0(x−2)(−x+1)≤0
Solve the inequality
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Evaluate
(x−2)(−x+1)≥0
Separate the inequality into 2 possible cases
{x−2≥0−x+1≥0{x−2≤0−x+1≤0
Solve the inequality
{x≥2−x+1≥0{x−2≤0−x+1≤0
Solve the inequality
{x≥2x≤1{x−2≤0−x+1≤0
Solve the inequality
{x≥2x≤1{x≤2−x+1≤0
Solve the inequality
{x≥2x≤1{x≤2x≥1
Find the intersection
x∈∅{x≤2x≥1
Find the intersection
x∈∅1≤x≤2
Find the union
1≤x≤2
{x≥31≤x≤2{x−3≤0(x−2)(−x+1)≤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≥31≤x≤2{x≤3(x−2)(−x+1)≤0
Solve the inequality
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Evaluate
(x−2)(−x+1)≤0
Separate the inequality into 2 possible cases
{x−2≥0−x+1≤0{x−2≤0−x+1≥0
Solve the inequality
{x≥2−x+1≤0{x−2≤0−x+1≥0
Solve the inequality
{x≥2x≥1{x−2≤0−x+1≥0
Solve the inequality
{x≥2x≥1{x≤2−x+1≥0
Solve the inequality
{x≥2x≥1{x≤2x≤1
Find the intersection
x≥2{x≤2x≤1
Find the intersection
x≥2x≤1
Find the union
x∈(−∞,1]∪[2,+∞)
{x≥31≤x≤2{x≤3x∈(−∞,1]∪[2,+∞)
Find the intersection
x∈∅{x≤3x∈(−∞,1]∪[2,+∞)
Find the intersection
x∈∅x∈(−∞,1]∪[2,3]
Find the union
x∈(−∞,1]∪[2,3]
{x∈(−∞,1]∪[2,3]x6>0{(x−3)(x−2)(−x+1)≤0x6<0
Solve the inequality
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Evaluate
x6>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,except when x6=0
x6=0
The only way a power can be 0 is when the base equals 0
x=0
Exclude the impossible values of x
x=0
{x∈(−∞,1]∪[2,3]x=0{(x−3)(x−2)(−x+1)≤0x6<0
Solve the inequality
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Evaluate
(x−3)(x−2)(−x+1)≤0
Separate the inequality into 2 possible cases
{x−3≥0(x−2)(−x+1)≤0{x−3≤0(x−2)(−x+1)≥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≥3(x−2)(−x+1)≤0{x−3≤0(x−2)(−x+1)≥0
Solve the inequality
More Steps

Evaluate
(x−2)(−x+1)≤0
Separate the inequality into 2 possible cases
{x−2≥0−x+1≤0{x−2≤0−x+1≥0
Solve the inequality
{x≥2−x+1≤0{x−2≤0−x+1≥0
Solve the inequality
{x≥2x≥1{x−2≤0−x+1≥0
Solve the inequality
{x≥2x≥1{x≤2−x+1≥0
Solve the inequality
{x≥2x≥1{x≤2x≤1
Find the intersection
x≥2{x≤2x≤1
Find the intersection
x≥2x≤1
Find the union
x∈(−∞,1]∪[2,+∞)
{x≥3x∈(−∞,1]∪[2,+∞){x−3≤0(x−2)(−x+1)≥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≥3x∈(−∞,1]∪[2,+∞){x≤3(x−2)(−x+1)≥0
Solve the inequality
More Steps

Evaluate
(x−2)(−x+1)≥0
Separate the inequality into 2 possible cases
{x−2≥0−x+1≥0{x−2≤0−x+1≤0
Solve the inequality
{x≥2−x+1≥0{x−2≤0−x+1≤0
Solve the inequality
{x≥2x≤1{x−2≤0−x+1≤0
Solve the inequality
{x≥2x≤1{x≤2−x+1≤0
Solve the inequality
{x≥2x≤1{x≤2x≥1
Find the intersection
x∈∅{x≤2x≥1
Find the intersection
x∈∅1≤x≤2
Find the union
1≤x≤2
{x≥3x∈(−∞,1]∪[2,+∞){x≤31≤x≤2
Find the intersection
x≥3{x≤31≤x≤2
Find the intersection
x≥31≤x≤2
Find the union
x∈[1,2]∪[3,+∞)
{x∈(−∞,1]∪[2,3]x=0{x∈[1,2]∪[3,+∞)x6<0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is false for any value of x
{x∈(−∞,1]∪[2,3]x=0{x∈[1,2]∪[3,+∞)x∈/R
Find the intersection
x∈(−∞,0)∪(0,1]∪[2,3]{x∈[1,2]∪[3,+∞)x∈/R
Find the intersection
x∈(−∞,0)∪(0,1]∪[2,3]x∈/R
Find the union
x∈(−∞,0)∪(0,1]∪[2,3]
Check if the solution is in the defined range
x∈(−∞,0)∪(0,1]∪[2,3],x=0
Solution
x∈(−∞,0)∪(0,1]∪[2,3]
Show Solution
