Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve for x
2−13+3<x<213+3
Alternative Form
x∈(2−13+3,213+3)
Evaluate
x2−3x−1<0
Rewrite the expression
x2−3x−1=0
Add or subtract both sides
x2−3x=1
Add the same value to both sides
x2−3x+49=1+49
Simplify the expression
(x−23)2=413
Take the root of both sides of the equation and remember to use both positive and negative roots
x−23=±413
Simplify the expression
x−23=±213
Separate the equation into 2 possible cases
x−23=213x−23=−213
Solve the equation
More Steps

Evaluate
x−23=213
Move the constant to the right-hand side and change its sign
x=213+23
Write all numerators above the common denominator
x=213+3
x=213+3x−23=−213
Solve the equation
More Steps

Evaluate
x−23=−213
Move the constant to the right-hand side and change its sign
x=−213+23
Write all numerators above the common denominator
x=2−13+3
x=213+3x=2−13+3
Determine the test intervals using the critical values
x<2−13+32−13+3<x<213+3x>213+3
Choose a value form each interval
x1=−1x2=2x3=4
To determine if x<2−13+3 is the solution to the inequality,test if the chosen value x=−1 satisfies the initial inequality
More Steps

Evaluate
(−1)2−3(−1)−1<0
Simplify
More Steps

Evaluate
(−1)2−3(−1)−1
Evaluate the power
1−3(−1)−1
Simplify
1+3−1
Apply the inverse property of addition
3
3<0
Check the inequality
false
x<2−13+3 is not a solutionx2=2x3=4
To determine if 2−13+3<x<213+3 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps

Evaluate
22−3×2−1<0
Simplify
More Steps

Evaluate
22−3×2−1
Multiply the numbers
22−6−1
Evaluate the power
4−6−1
Subtract the numbers
−3
−3<0
Check the inequality
true
x<2−13+3 is not a solution2−13+3<x<213+3 is the solutionx3=4
To determine if x>213+3 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
More Steps

Evaluate
42−3×4−1<0
Simplify
More Steps

Evaluate
42−3×4−1
Multiply the numbers
42−12−1
Evaluate the power
16−12−1
Subtract the numbers
3
3<0
Check the inequality
false
x<2−13+3 is not a solution2−13+3<x<213+3 is the solutionx>213+3 is not a solution
Solution
2−13+3<x<213+3
Alternative Form
x∈(2−13+3,213+3)
Show Solution
