Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
0≤a<93127+56
Alternative Form
a∈[0,93127+56)
Evaluate
186a−2−2106a<0
Find the domain
More Steps
Evaluate
106a≥0
Rewrite the expression
a≥0
186a−2−2106a<0,a≥0
Divide the terms
More Steps
Evaluate
186a−2−2106a
Factor
182(3a−1−106a)
Reduce the fraction
93a−1−106a
93a−1−106a<0
Multiply both sides of the inequality by 9
93a−1−106a×9<0×9
Multiply the terms
3a−1−106a<0×9
Multiply the terms
3a−1−106a<0
Change the signs on both sides of the inequality and flip the inequality sign
106a−3a+1>0
Move the expression to the right side
106a>3a−1
Separate the inequality into 2 possible cases
106a>3a−1,3a−1≥0106a>3a−1,3a−1<0
Solve the inequality
More Steps
Solve the inequality
106a>3a−1
Square both sides of the inequality
106a>(3a−1)2
Expand the expression
106a>9a2−6a+1
Move the expression to the left side
106a−(9a2−6a+1)>0
Subtract the terms
More Steps
Evaluate
106a−(9a2−6a+1)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
106a−9a2+6a−1
Add the terms
112a−9a2−1
112a−9a2−1>0
Move the constant to the right side
112a−9a2>0−(−1)
Add the terms
112a−9a2>1
Evaluate
a2−9112a<−91
Add the same value to both sides
a2−9112a+813136<−91+813136
Evaluate
a2−9112a+813136<813127
Evaluate
(a−956)2<813127
Take the 2-th root on both sides of the inequality
(a−956)2<813127
Calculate
a−956<93127
Separate the inequality into 2 possible cases
{a−956<93127a−956>−93127
Calculate
More Steps
Evaluate
a−956<93127
Move the constant to the right side
a<93127+956
Write all numerators above the common denominator
a<93127+56
{a<93127+56a−956>−93127
Calculate
More Steps
Evaluate
a−956>−93127
Move the constant to the right side
a>−93127+956
Write all numerators above the common denominator
a>9−3127+56
{a<93127+56a>9−3127+56
Find the intersection
9−3127+56<a<93127+56
9−3127+56<a<93127+56,3a−1≥0106a>3a−1,3a−1<0
Solve the inequality
More Steps
Evaluate
3a−1≥0
Move the constant to the right side
3a≥0+1
Removing 0 doesn't change the value,so remove it from the expression
3a≥1
Divide both sides
33a≥31
Divide the numbers
a≥31
9−3127+56<a<93127+56,a≥31106a>3a−1,3a−1<0
Since the left-hand side is always positive or 0,and the right-hand side is always negative,the statement is true for any value of a
9−3127+56<a<93127+56,a≥31a∈R,3a−1<0
Solve the inequality
More Steps
Evaluate
3a−1<0
Move the constant to the right side
3a<0+1
Removing 0 doesn't change the value,so remove it from the expression
3a<1
Divide both sides
33a<31
Divide the numbers
a<31
9−3127+56<a<93127+56,a≥31a∈R,a<31
Find the intersection
31≤a<93127+56a∈R,a<31
Find the intersection
31≤a<93127+56a<31
Find the union
a<93127+56
Check if the solution is in the defined range
a<93127+56,a≥0
Solution
0≤a<93127+56
Alternative Form
a∈[0,93127+56)
Show Solution
