Question
Solve the inequality
Solve the inequality by testing the values in the interval
Solve the inequality by separating into cases
x∈(4−313+29,11)∪(4313+29,+∞)
Evaluate
x÷(x−11)−5<(2x−11)×1
Find the domain
More Steps

Evaluate
x−11=0
Move the constant to the right side
x=0+11
Removing 0 doesn't change the value,so remove it from the expression
x=11
x÷(x−11)−5<(2x−11)×1,x=11
Rewrite the expression
x−11x−5<(2x−11)×1
Any expression multiplied by 1 remains the same
x−11x−5<2x−11
Move the expression to the left side
x−11x−5−(2x−11)<0
Subtract the terms
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Evaluate
x−11x−5−(2x−11)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x−11x−5−2x+11
Add the numbers
x−11x+6−2x
Reduce fractions to a common denominator
x−11x+x−116(x−11)−x−112x(x−11)
Write all numerators above the common denominator
x−11x+6(x−11)−2x(x−11)
Multiply the terms
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Evaluate
6(x−11)
Apply the distributive property
6x−6×11
Multiply the numbers
6x−66
x−11x+6x−66−2x(x−11)
Multiply the terms
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Evaluate
2x(x−11)
Multiply the terms
(2x−22)x
Apply the distributive property
2x×x−22x
Multiply the terms
2x2−22x
x−11x+6x−66−(2x2−22x)
Calculate the sum or difference
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Evaluate
x+6x−66−(2x2−22x)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
x+6x−66−2x2+22x
Add the terms
29x−66−2x2
x−1129x−66−2x2
x−1129x−66−2x2<0
Set the numerator and denominator of x−1129x−66−2x2 equal to 0 to find the values of x where sign changes may occur
29x−66−2x2=0x−11=0
Calculate
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Evaluate
29x−66−2x2=0
Add or subtract both sides
29x−2x2=66
Divide both sides
−229x−2x2=−266
Evaluate
−229x+x2=−33
Add the same value to both sides
−229x+x2+16841=−33+16841
Simplify the expression
(x−429)2=16313
Take the root of both sides of the equation and remember to use both positive and negative roots
x−429=±16313
Simplify the expression
x−429=±4313
Separate the equation into 2 possible cases
x−429=4313x−429=−4313
Solve the equation
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Evaluate
x−429=4313
Move the constant to the right-hand side and change its sign
x=4313+429
Write all numerators above the common denominator
x=4313+29
x=4313+29x−429=−4313
Solve the equation
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Evaluate
x−429=−4313
Move the constant to the right-hand side and change its sign
x=−4313+429
Write all numerators above the common denominator
x=4−313+29
x=4313+29x=4−313+29
x=4313+29x=4−313+29x−11=0
Calculate
More Steps

Evaluate
x−11=0
Move the constant to the right-hand side and change its sign
x=0+11
Removing 0 doesn't change the value,so remove it from the expression
x=11
x=4313+29x=4−313+29x=11
Determine the test intervals using the critical values
x<4−313+294−313+29<x<1111<x<4313+29x>4313+29
Choose a value form each interval
x1=2x2=7x3=8313+73x4=13
To determine if x<4−313+29 is the solution to the inequality,test if the chosen value x=2 satisfies the initial inequality
More Steps

Evaluate
2−112−5<2×2−11
Simplify
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Evaluate
2−112−5
Subtract the numbers
−92−5
Use b−a=−ba=−ba to rewrite the fraction
−92−5
Reduce fractions to a common denominator
−92−95×9
Write all numerators above the common denominator
9−2−5×9
Multiply the numbers
9−2−45
Subtract the numbers
9−47
Use b−a=−ba=−ba to rewrite the fraction
−947
−947<2×2−11
Simplify
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Evaluate
2×2−11
Multiply the numbers
4−11
Subtract the numbers
−7
−947<−7
Calculate
−5.2˙<−7
Check the inequality
false
x<4−313+29 is not a solutionx2=7x3=8313+73x4=13
To determine if 4−313+29<x<11 is the solution to the inequality,test if the chosen value x=7 satisfies the initial inequality
More Steps

Evaluate
7−117−5<2×7−11
Simplify
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Evaluate
7−117−5
Subtract the numbers
−47−5
Use b−a=−ba=−ba to rewrite the fraction
−47−5
Reduce fractions to a common denominator
−47−45×4
Write all numerators above the common denominator
4−7−5×4
Multiply the numbers
4−7−20
Subtract the numbers
4−27
Use b−a=−ba=−ba to rewrite the fraction
−427
−427<2×7−11
Simplify
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Evaluate
2×7−11
Multiply the numbers
14−11
Subtract the numbers
3
−427<3
Calculate
−6.75<3
Check the inequality
true
x<4−313+29 is not a solution4−313+29<x<11 is the solutionx3=8313+73x4=13
To determine if 11<x<4313+29 is the solution to the inequality,test if the chosen value x=8313+73 satisfies the initial inequality
More Steps

Evaluate
8313+73−118313+73−5<2×8313+73−11
Simplify
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Evaluate
8313+73−118313+73−5
Subtract the numbers
8313−158313+73−5
Divide the terms
313−15313+73−5
Calculate
16+313−5
Subtract the numbers
11+313
11+313<2×8313+73−11
Simplify
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Evaluate
2×8313+73−11
Multiply the numbers
4313+73−11
Reduce fractions to a common denominator
4313+73−411×4
Write all numerators above the common denominator
4313+73−11×4
Multiply the numbers
4313+73−44
Subtract the numbers
4313+29
11+313<4313+29
Calculate
28.691806<4313+29
Calculate
28.691806<11.672952
Check the inequality
false
x<4−313+29 is not a solution4−313+29<x<11 is the solution11<x<4313+29 is not a solutionx4=13
To determine if x>4313+29 is the solution to the inequality,test if the chosen value x=13 satisfies the initial inequality
More Steps

Evaluate
13−1113−5<2×13−11
Simplify
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Evaluate
13−1113−5
Subtract the numbers
213−5
Reduce fractions to a common denominator
213−25×2
Write all numerators above the common denominator
213−5×2
Multiply the numbers
213−10
Subtract the numbers
23
23<2×13−11
Simplify
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Evaluate
2×13−11
Multiply the numbers
26−11
Subtract the numbers
15
23<15
Calculate
1.5<15
Check the inequality
true
x<4−313+29 is not a solution4−313+29<x<11 is the solution11<x<4313+29 is not a solutionx>4313+29 is the solution
The original inequality is a strict inequality,so does not include the critical value ,the final solution is x∈(4−313+29,11)∪(4313+29,+∞)
x∈(4−313+29,11)∪(4313+29,+∞)
Check if the solution is in the defined range
x∈(4−313+29,11)∪(4313+29,+∞),x=11
Solution
x∈(4−313+29,11)∪(4313+29,+∞)
Show Solution
