Question :
x^2−5x+6le 0
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
Solve for x
2≤x≤3
Alternative Form
x∈[2,3]
Evaluate
x2−5x+6≤0
Rewrite the expression
x2−5x+6=0
Factor the expression
More Steps
Evaluate
x2−5x+6
Rewrite the expression
x2+(−2−3)x+6
Calculate
x2−2x−3x+6
Rewrite the expression
x×x−x×2−3x+3×2
Factor out x from the expression
x(x−2)−3x+3×2
Factor out −3 from the expression
x(x−2)−3(x−2)
Factor out x−2 from the expression
(x−3)(x−2)
(x−3)(x−2)=0
When the product of factors equals 0,at least one factor is 0
x−3=0x−2=0
Solve the equation for x
More Steps
Evaluate
x−3=0
Move the constant to the right-hand side and change its sign
x=0+3
Removing 0 doesn't change the value,so remove it from the expression
x=3
x=3x−2=0
Solve the equation for x
More Steps
Evaluate
x−2=0
Move the constant to the right-hand side and change its sign
x=0+2
Removing 0 doesn't change the value,so remove it from the expression
x=2
x=3x=2
Determine the test intervals using the critical values
x<22<x<3x>3
Choose a value form each interval
x1=1x2=25x3=4
To determine if x<2 is the solution to the inequality,test if the chosen value x=1 satisfies the initial inequality
More Steps
Evaluate
12−5×1+6≤0
Simplify
More Steps
Evaluate
12−5×1+6
1 raised to any power equals to 1
1−5×1+6
Any expression multiplied by 1 remains the same
1−5+6
Calculate the sum or difference
2
2≤0
Check the inequality
false
x<2 is not a solutionx2=25x3=4
To determine if 2<x<3 is the solution to the inequality,test if the chosen value x=25 satisfies the initial inequality
More Steps
Evaluate
(25)2−5×25+6≤0
Simplify
More Steps
Evaluate
(25)2−5×25+6
Multiply the numbers
(25)2−225+6
Evaluate the power
425−225+6
Subtract the numbers
−425+6
Reduce fractions to a common denominator
−425+46×4
Write all numerators above the common denominator
4−25+6×4
Multiply the numbers
4−25+24
Add the numbers
4−1
Use b−a=−ba=−ba to rewrite the fraction
−41
−41≤0
Calculate
−0.25≤0
Check the inequality
true
x<2 is not a solution2<x<3 is the solutionx3=4
To determine if x>3 is the solution to the inequality,test if the chosen value x=4 satisfies the initial inequality
More Steps
Evaluate
42−5×4+6≤0
Simplify
More Steps
Evaluate
42−5×4+6
Multiply the numbers
42−20+6
Evaluate the power
16−20+6
Calculate the sum or difference
2
2≤0
Check the inequality
false
x<2 is not a solution2<x<3 is the solutionx>3 is not a solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
2≤x≤3 is the solution
Solution
2≤x≤3
Alternative Form
x∈[2,3]
Show Solution
