Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(331,12)∪(12,341)
Evaluate
x−122>56
Find the domain
More Steps
Evaluate
x−12=0
Move the constant to the right side
x=0+12
Removing 0 doesn't change the value,so remove it from the expression
x=12
x−122>56,x=12
Simplify
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Evaluate
x−122
Rewrite the expression
2×x−121
Rewrite the expression
2x−121
2x−121>56
Calculate
5∣x−12∣10−6∣x−12∣>0
Separate the inequality into 2 possible cases
{10−6∣x−12∣>05∣x−12∣>0{10−6∣x−12∣<05∣x−12∣<0
Solve the inequality
More Steps
Evaluate
10−6∣x−12∣>0
Rewrite the expression
−6∣x−12∣>−10
Change the signs on both sides of the inequality and flip the inequality sign
6∣x−12∣<10
Divide both sides
66∣x−12∣<610
Divide the numbers
∣x−12∣<610
Cancel out the common factor 2
∣x−12∣<35
Separate the inequality into 2 possible cases
{x−12<35x−12>−35
Solve the inequality for x
More Steps
Evaluate
x−12<35
Move the constant to the right side
x<35+12
Add the numbers
x<341
{x<341x−12>−35
Solve the inequality for x
More Steps
Evaluate
x−12>−35
Move the constant to the right side
x>−35+12
Add the numbers
x>331
{x<341x>331
Find the intersection
331<x<341
{331<x<3415∣x−12∣>0{10−6∣x−12∣<05∣x−12∣<0
Solve the inequality
More Steps
Evaluate
5∣x−12∣>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 5∣x−12∣=0
5∣x−12∣=0
Rewrite the expression
∣x−12∣=0
Rewrite the expression
x−12=0
Move the constant to the right-hand side and change its sign
x=0+12
Removing 0 doesn't change the value,so remove it from the expression
x=12
Exclude the impossible values of x
x=12
{331<x<341x=12{10−6∣x−12∣<05∣x−12∣<0
Solve the inequality
More Steps
Evaluate
10−6∣x−12∣<0
Rewrite the expression
−6∣x−12∣<−10
Change the signs on both sides of the inequality and flip the inequality sign
6∣x−12∣>10
Divide both sides
66∣x−12∣>610
Divide the numbers
∣x−12∣>610
Cancel out the common factor 2
∣x−12∣>35
Separate the inequality into 2 possible cases
x−12>35x−12<−35
Solve the inequality for x
More Steps
Evaluate
x−12>35
Move the constant to the right side
x>35+12
Add the numbers
x>341
x>341x−12<−35
Solve the inequality for x
More Steps
Evaluate
x−12<−35
Move the constant to the right side
x<−35+12
Add the numbers
x<331
x>341x<331
Find the union
x∈(−∞,331)∪(341,+∞)
{331<x<341x=12{x∈(−∞,331)∪(341,+∞)5∣x−12∣<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
{331<x<341x=12{x∈(−∞,331)∪(341,+∞)x∈/R
Find the intersection
x∈(331,12)∪(12,341){x∈(−∞,331)∪(341,+∞)x∈/R
Find the intersection
x∈(331,12)∪(12,341)x∈/R
Find the union
x∈(331,12)∪(12,341)
Check if the solution is in the defined range
x∈(331,12)∪(12,341),x=12
Solution
x∈(331,12)∪(12,341)
Show Solution
