Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve for n
2−42481+1<n<242481+1
Alternative Form
n∈(2−42481+1,242481+1)
Evaluate
n2−n×1<10620
Any expression multiplied by 1 remains the same
n2−n<10620
Move the expression to the left side
n2−n−10620<0
Rewrite the expression
n2−n−10620=0
Add or subtract both sides
n2−n=10620
Add the same value to both sides
n2−n+41=10620+41
Simplify the expression
(n−21)2=442481
Take the root of both sides of the equation and remember to use both positive and negative roots
n−21=±442481
Simplify the expression
n−21=±242481
Separate the equation into 2 possible cases
n−21=242481n−21=−242481
Solve the equation
More Steps

Evaluate
n−21=242481
Move the constant to the right-hand side and change its sign
n=242481+21
Write all numerators above the common denominator
n=242481+1
n=242481+1n−21=−242481
Solve the equation
More Steps

Evaluate
n−21=−242481
Move the constant to the right-hand side and change its sign
n=−242481+21
Write all numerators above the common denominator
n=2−42481+1
n=242481+1n=2−42481+1
Determine the test intervals using the critical values
n<2−42481+12−42481+1<n<242481+1n>242481+1
Choose a value form each interval
n1=−104n2=1n3=105
To determine if n<2−42481+1 is the solution to the inequality,test if the chosen value n=−104 satisfies the initial inequality
More Steps

Evaluate
(−104)2−(−104)<10620
Subtract the terms
More Steps

Simplify
(−104)2−(−104)
Rewrite the expression
1042−(−104)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
1042+104
Evaluate the power
10816+104
Add the numbers
10920
10920<10620
Check the inequality
false
n<2−42481+1 is not a solutionn2=1n3=105
To determine if 2−42481+1<n<242481+1 is the solution to the inequality,test if the chosen value n=1 satisfies the initial inequality
More Steps

Evaluate
12−1<10620
Simplify
More Steps

Evaluate
12−1
1 raised to any power equals to 1
1−1
Subtract the terms
0
0<10620
Check the inequality
true
n<2−42481+1 is not a solution2−42481+1<n<242481+1 is the solutionn3=105
To determine if n>242481+1 is the solution to the inequality,test if the chosen value n=105 satisfies the initial inequality
More Steps

Evaluate
1052−105<10620
Subtract the numbers
More Steps

Evaluate
1052−105
Evaluate the power
11025−105
Subtract the numbers
10920
10920<10620
Check the inequality
false
n<2−42481+1 is not a solution2−42481+1<n<242481+1 is the solutionn>242481+1 is not a solution
Solution
2−42481+1<n<242481+1
Alternative Form
n∈(2−42481+1,242481+1)
Show Solution
