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
Solve for x
1<x<2+1
Alternative Form
x∈(1,2+1)
Evaluate
(x−1)2<2×(x−1)
Multiply the terms
More Steps
Evaluate
2×(x−1)
Use the the distributive property to expand the expression
2×x+2×(−1)
Multiply the numbers
2×x−2
(x−1)2<2×x−2
Move the expression to the left side
(x−1)2−(2×x−2)<0
Subtract the terms
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Evaluate
(x−1)2−(2×x−2)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
(x−1)2−2×x+2
Expand the expression
x2−2x+1−2×x+2
x2−2x+1−2×x+2<0
Rewrite the expression
x2−2x+1−2×x+2=0
Collect like terms by calculating the sum or difference of their coefficients
x2+(−2−2)x+1+2=0
Expand the expression
x2−2x−2×x+1+2=0
Factor the expression
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Evaluate
x2−2x−2×x+1+2
Calculate
x2−x−2×x+2−x+1
Rewrite the expression
x×x−x−2×x+2−x+1
Factor out x from the expression
x(x−1)−2×x+2−x+1
Factor out −2 from the expression
x(x−1)−2×(x−1)−x+1
Factor out −1 from the expression
x(x−1)−2×(x−1)−(x−1)
Factor out x−1 from the expression
(x−2−1)(x−1)
(x−2−1)(x−1)=0
When the product of factors equals 0,at least one factor is 0
x−2−1=0x−1=0
Solve the equation for x
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Evaluate
x−2−1=0
Move the constant to the right-hand side and change its sign
x=0+2+1
Removing 0 doesn't change the value,so remove it from the expression
x=2+1
x=2+1x−1=0
Solve the equation for x
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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=2+1x=1
Determine the test intervals using the critical values
x<11<x<2+1x>2+1
Choose a value form each interval
x1=0x2=2x3=3
To determine if x<1 is the solution to the inequality,test if the chosen value x=0 satisfies the initial inequality
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Evaluate
(0−1)2<2×0−2
Any expression multiplied by 0 equals 0
(0−1)2<0−2
Simplify
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Evaluate
(0−1)2
Removing 0 doesn't change the value,so remove it from the expression
(−1)2
Evaluate the power
1
1<0−2
Removing 0 doesn't change the value,so remove it from the expression
1<−2
Calculate
1<−1.414214
Check the inequality
false
x<1 is not a solutionx2=2x3=3
To determine if 1<x<2+1 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps
Evaluate
(2−1)2<2×2−2
Simplify
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Evaluate
(2−1)2
Subtract the numbers
12
1 raised to any power equals to 1
1
1<2×2−2
Simplify
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Evaluate
2×2−2
Multiply the numbers
22−2
Rewrite the expression
(2−1)2
Subtract the numbers
2
1<2
Calculate
1<1.414214
Check the inequality
true
x<1 is not a solution1<x<2+1 is the solutionx3=3
To determine if x>2+1 is the solution to the inequality,test if the chosen value x=3 satisfies the initial inequality
More Steps
Evaluate
(3−1)2<2×3−2
Subtract the numbers
22<2×3−2
Simplify
More Steps
Evaluate
2×3−2
Multiply the numbers
32−2
Rewrite the expression
(3−1)2
Subtract the numbers
22
22<22
Calculate
4<22
Calculate
4<2.828427
Check the inequality
false
x<1 is not a solution1<x<2+1 is the solutionx>2+1 is not a solution
Solution
1<x<2+1
Alternative Form
x∈(1,2+1)
Show Solution
