Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(27,4)∪(4,29)
Evaluate
x−42>4
Find the domain
More Steps
Evaluate
x−4=0
Move the constant to the right side
x=0+4
Removing 0 doesn't change the value,so remove it from the expression
x=4
x−42>4,x=4
Simplify
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Evaluate
x−42
Rewrite the expression
2×x−41
Rewrite the expression
2x−41
2x−41>4
Calculate
∣x−4∣2−4∣x−4∣>0
Separate the inequality into 2 possible cases
{2−4∣x−4∣>0∣x−4∣>0{2−4∣x−4∣<0∣x−4∣<0
Solve the inequality
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Evaluate
2−4∣x−4∣>0
Rewrite the expression
−4∣x−4∣>−2
Change the signs on both sides of the inequality and flip the inequality sign
4∣x−4∣<2
Divide both sides
44∣x−4∣<42
Divide the numbers
∣x−4∣<42
Cancel out the common factor 2
∣x−4∣<21
Separate the inequality into 2 possible cases
{x−4<21x−4>−21
Solve the inequality for x
More Steps
Evaluate
x−4<21
Move the constant to the right side
x<21+4
Add the numbers
x<29
{x<29x−4>−21
Solve the inequality for x
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Evaluate
x−4>−21
Move the constant to the right side
x>−21+4
Add the numbers
x>27
{x<29x>27
Find the intersection
27<x<29
{27<x<29∣x−4∣>0{2−4∣x−4∣<0∣x−4∣<0
Solve the inequality
More Steps
Evaluate
∣x−4∣>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 ∣x−4∣=0
∣x−4∣=0
Rewrite the expression
x−4=0
Move the constant to the right-hand side and change its sign
x=0+4
Removing 0 doesn't change the value,so remove it from the expression
x=4
Exclude the impossible values of x
x=4
{27<x<29x=4{2−4∣x−4∣<0∣x−4∣<0
Solve the inequality
More Steps
Evaluate
2−4∣x−4∣<0
Rewrite the expression
−4∣x−4∣<−2
Change the signs on both sides of the inequality and flip the inequality sign
4∣x−4∣>2
Divide both sides
44∣x−4∣>42
Divide the numbers
∣x−4∣>42
Cancel out the common factor 2
∣x−4∣>21
Separate the inequality into 2 possible cases
x−4>21x−4<−21
Solve the inequality for x
More Steps
Evaluate
x−4>21
Move the constant to the right side
x>21+4
Add the numbers
x>29
x>29x−4<−21
Solve the inequality for x
More Steps
Evaluate
x−4<−21
Move the constant to the right side
x<−21+4
Add the numbers
x<27
x>29x<27
Find the union
x∈(−∞,27)∪(29,+∞)
{27<x<29x=4{x∈(−∞,27)∪(29,+∞)∣x−4∣<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
{27<x<29x=4{x∈(−∞,27)∪(29,+∞)x∈/R
Find the intersection
x∈(27,4)∪(4,29){x∈(−∞,27)∪(29,+∞)x∈/R
Find the intersection
x∈(27,4)∪(4,29)x∈/R
Find the union
x∈(27,4)∪(4,29)
Check if the solution is in the defined range
x∈(27,4)∪(4,29),x=4
Solution
x∈(27,4)∪(4,29)
Show Solution
