Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
Solve for x
3≤x≤311
Alternative Form
x∈[3,311]
Evaluate
42x2−3x×1≤3x−633
Simplify
More Steps

Evaluate
42x2−3x×1
Any expression multiplied by 1 remains the same
42x2−3x
Cancel out the common factor 2
2x2−3x
2x2−3x≤3x−633
Cancel out the common factor 3
2x2−3x≤3x−211
Multiply both sides of the inequality by 6
(2x2−3x)×6≤(3x−211)×6
Multiply the terms
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Multiply the terms
(2x2−3x)×6
Apply the distributive property
2x2×6−3x×6
Reduce the fraction
x2×3−x×2
Multiply the terms
3x2−2x
3x2−2x≤(3x−211)×6
Multiply the terms
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Multiply the terms
(3x−211)×6
Apply the distributive property
3x×6−211×6
Reduce the fraction
3x×6−11×3
Multiply the terms
18x−33
3x2−2x≤18x−33
Move the expression to the left side
3x2−2x−(18x−33)≤0
Subtract the terms
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Evaluate
3x2−2x−(18x−33)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
3x2−2x−18x+33
Subtract the terms
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Evaluate
−2x−18x
Collect like terms by calculating the sum or difference of their coefficients
(−2−18)x
Subtract the numbers
−20x
3x2−20x+33
3x2−20x+33≤0
Rewrite the expression
3x2−20x+33=0
Factor the expression
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Evaluate
3x2−20x+33
Rewrite the expression
3x2+(−11−9)x+33
Calculate
3x2−11x−9x+33
Rewrite the expression
x×3x−x×11−3×3x+3×11
Factor out x from the expression
x(3x−11)−3×3x+3×11
Factor out −3 from the expression
x(3x−11)−3(3x−11)
Factor out 3x−11 from the expression
(x−3)(3x−11)
(x−3)(3x−11)=0
When the product of factors equals 0,at least one factor is 0
x−3=03x−11=0
Solve the equation for x
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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=33x−11=0
Solve the equation for x
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Evaluate
3x−11=0
Move the constant to the right-hand side and change its sign
3x=0+11
Removing 0 doesn't change the value,so remove it from the expression
3x=11
Divide both sides
33x=311
Divide the numbers
x=311
x=3x=311
Determine the test intervals using the critical values
x<33<x<311x>311
Choose a value form each interval
x1=2x2=310x3=5
To determine if x<3 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps

Evaluate
3×22−2×2≤18×2−33
Simplify
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Evaluate
3×22−2×2
Multiply the terms
12−2×2
Multiply the numbers
12−4
Subtract the numbers
8
8≤18×2−33
Simplify
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Evaluate
18×2−33
Multiply the numbers
36−33
Subtract the numbers
3
8≤3
Check the inequality
false
x<3 is not a solutionx2=310x3=5
To determine if 3<x<311 is the solution to the inequality,test if the chosen value x=310 satisfies the initial inequality
More Steps

Evaluate
3(310)2−2×310≤18×310−33
Simplify
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Evaluate
3(310)2−2×310
Multiply the terms
3100−2×310
Multiply the numbers
3100−320
Write all numerators above the common denominator
3100−20
Subtract the numbers
380
380≤18×310−33
Simplify
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Evaluate
18×310−33
Multiply the numbers
60−33
Subtract the numbers
27
380≤27
Calculate
26.6˙≤27
Check the inequality
true
x<3 is not a solution3<x<311 is the solutionx3=5
To determine if x>311 is the solution to the inequality,test if the chosen value x=5 satisfies the initial inequality
More Steps

Evaluate
3×52−2×5≤18×5−33
Simplify
More Steps

Evaluate
3×52−2×5
Multiply the terms
75−2×5
Multiply the numbers
75−10
Subtract the numbers
65
65≤18×5−33
Simplify
More Steps

Evaluate
18×5−33
Multiply the numbers
90−33
Subtract the numbers
57
65≤57
Check the inequality
false
x<3 is not a solution3<x<311 is the solutionx>311 is not a solution
The original inequality is a nonstrict inequality,so include the critical value in the solution
3≤x≤311 is the solution
Solution
3≤x≤311
Alternative Form
x∈[3,311]
Show Solution
