Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
1≤a≤35
Alternative Form
a∈[1,35]
Evaluate
0≤186a−2−210−6a≤21
Find the domain
More Steps
Evaluate
10−6a≥0
Move the constant to the right side
−6a≥0−10
Removing 0 doesn't change the value,so remove it from the expression
−6a≥−10
Change the signs on both sides of the inequality and flip the inequality sign
6a≤10
Divide both sides
66a≤610
Divide the numbers
a≤610
Cancel out the common factor 2
a≤35
0≤186a−2−210−6a≤21,a≤35
Separate into two inequalities
{0≤186a−2−210−6a186a−2−210−6a≤21
Solve the inequality
More Steps
Evaluate
0≤186a−2−210−6a
Divide the terms
More Steps
Evaluate
186a−2−210−6a
Factor
182(3a−1−10−6a)
Reduce the fraction
93a−1−10−6a
0≤93a−1−10−6a
Swap the sides of the inequality
93a−1−10−6a≥0
Simplify
3a−1−10−6a≥0
Change the signs on both sides of the inequality and flip the inequality sign
10−6a−3a+1≤0
Move the expression to the right side
10−6a≤3a−1
Separate the inequality into 2 possible cases
10−6a≤3a−1,3a−1≥010−6a≤3a−1,3a−1<0
Solve the inequality
More Steps
Solve the inequality
10−6a≤3a−1
Square both sides of the inequality
10−6a≤(3a−1)2
Move the expression to the left side
10−6a−(3a−1)2≤0
Calculate
9−9a2≤0
Rewrite the expression
−9a2≤−9
Change the signs on both sides of the inequality and flip the inequality sign
9a2≥9
Divide both sides
99a2≥99
Divide the numbers
a2≥99
Divide the numbers
a2≥1
Take the 2-th root on both sides of the inequality
a2≥1
Calculate
∣a∣≥1
Separate the inequality into 2 possible cases
a≥1a≤−1
Find the union
a∈(−∞,−1]∪[1,+∞)
a∈(−∞,−1]∪[1,+∞),3a−1≥010−6a≤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
a∈(−∞,−1]∪[1,+∞),a≥3110−6a≤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 false for any value of a
a∈(−∞,−1]∪[1,+∞),a≥31a∈∅,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
a∈(−∞,−1]∪[1,+∞),a≥31a∈∅,a<31
Find the intersection
a≥1a∈∅,a<31
Find the intersection
a≥1a∈∅
Find the union
a≥1
{a≥1186a−2−210−6a≤21
Solve the inequality
More Steps
Evaluate
186a−2−210−6a≤21
Divide the terms
More Steps
Evaluate
186a−2−210−6a
Factor
182(3a−1−10−6a)
Reduce the fraction
93a−1−10−6a
93a−1−10−6a≤21
Multiply both sides of the equation by 9
93a−1−10−6a×9≤21×9
Multiply the terms
3a−1−10−6a≤29
Move the expression to the left side
3a−1−10−6a−29≤0
Subtract the numbers
More Steps
Evaluate
−1−29
Reduce fractions to a common denominator
−22−29
Write all numerators above the common denominator
2−2−9
Subtract the numbers
2−11
Use b−a=−ba=−ba to rewrite the fraction
−211
3a−211−10−6a≤0
Change the signs on both sides of the inequality and flip the inequality sign
10−6a−3a+211≥0
Move the expression to the right side
10−6a≥3a−211
Separate the inequality into 2 possible cases
10−6a≥3a−211,3a−211≥010−6a≥3a−211,3a−211<0
Solve the inequality
More Steps
Solve the inequality
10−6a≥3a−211
Square both sides of the inequality
10−6a≥(3a−211)2
Move the expression to the left side
10−6a−(3a−211)2≥0
Calculate
−481+27a−9a2≥0
Move the constant to the right side
27a−9a2≥0−(−481)
Add the terms
27a−9a2≥481
Evaluate
a2−3a≤−49
Add the same value to both sides
a2−3a+49≤−49+49
Evaluate
a2−3a+49≤0
Evaluate
(a−23)2≤0
Calculate
(a−23)2=0
The only way a power can be 0 is when the base equals 0
a−23=0
Move the constant to the right-hand side and change its sign
a=0+23
Add the terms
a=23
a=23,3a−211≥010−6a≥3a−211,3a−211<0
Solve the inequality
More Steps
Evaluate
3a−211≥0
Move the constant to the right side
3a≥0+211
Removing 0 doesn't change the value,so remove it from the expression
3a≥211
Multiply by the reciprocal
3a×31≥211×31
Multiply
a≥211×31
Multiply
a≥611
a=23,a≥61110−6a≥3a−211,3a−211<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
a=23,a≥611a∈R,3a−211<0
Solve the inequality
More Steps
Evaluate
3a−211<0
Move the constant to the right side
3a<0+211
Removing 0 doesn't change the value,so remove it from the expression
3a<211
Multiply by the reciprocal
3a×31<211×31
Multiply
a<211×31
Multiply
a<611
a=23,a≥611a∈R,a<611
Find the intersection
a∈∅a∈R,a<611
Find the intersection
a∈∅a<611
Find the union
a<611
{a≥1a<611
Find the intersection
1≤a<611
Check if the solution is in the defined range
1≤a<611,a≤35
Solution
1≤a≤35
Alternative Form
a∈[1,35]
Show Solution
