Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
1<x<7
Alternative Form
x∈(1,7)
Evaluate
x5x<x<7
Find the domain
More Steps
Evaluate
x5=0
The only way a power can not be 0 is when the base not equals 0
x=0
x5x<x<7,x=0
Separate into two inequalities
{x5x<xx<7
Solve the inequality
More Steps
Evaluate
x5x<x
Divide the terms
More Steps
Evaluate
x5x
Use the product rule aman=an−m to simplify the expression
x5−11
Reduce the fraction
x41
x41<x
Move the expression to the left side
x41−x<0
Subtract the terms
More Steps
Evaluate
x41−x
Reduce fractions to a common denominator
x41−x4x×x4
Write all numerators above the common denominator
x41−x×x4
Multiply the terms
x41−x5
x41−x5<0
Separate the inequality into 2 possible cases
{1−x5>0x4<0{1−x5<0x4>0
Solve the inequality
More Steps
Evaluate
1−x5>0
Rewrite the expression
−x5>−1
Change the signs on both sides of the inequality and flip the inequality sign
x5<1
Take the 5-th root on both sides of the equation
5x5<51
Calculate
x<51
Simplify the root
x<1
{x<1x4<0{1−x5<0x4>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<1x∈/R{1−x5<0x4>0
Solve the inequality
More Steps
Evaluate
1−x5<0
Rewrite the expression
−x5<−1
Change the signs on both sides of the inequality and flip the inequality sign
x5>1
Take the 5-th root on both sides of the equation
5x5>51
Calculate
x>51
Simplify the root
x>1
{x<1x∈/R{x>1x4>0
Solve the inequality
More Steps
Evaluate
x4>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 x4=0
x4=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<1x∈/R{x>1x=0
Find the intersection
x∈/R{x>1x=0
Find the intersection
x∈/Rx>1
Find the union
x>1
{x>1x<7
Find the intersection
1<x<7
Check if the solution is in the defined range
1<x<7,x=0
Solution
1<x<7
Alternative Form
x∈(1,7)
Show Solution
