Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
a∈[−22+4,34)∪(4,22+4]
Evaluate
2∣2a−4∣−8a2−4(2a−4)2>0
Find the domain
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Evaluate
8a2−4(2a−4)2≥0
Calculate
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Evaluate
8a2−4(2a−4)2
Expand the expression
8a2−16a2+64a−64
Subtract the terms
−8a2+64a−64
−8a2+64a−64≥0
Move the constant to the right side
−8a2+64a≥0−(−64)
Add the terms
−8a2+64a≥64
Evaluate
a2−8a≤−8
Add the same value to both sides
a2−8a+16≤−8+16
Evaluate
a2−8a+16≤8
Evaluate
(a−4)2≤8
Take the 2-th root on both sides of the inequality
(a−4)2≤8
Calculate
∣a−4∣≤22
Separate the inequality into 2 possible cases
{a−4≤22a−4≥−22
Calculate
{a≤22+4a−4≥−22
Calculate
{a≤22+4a≥−22+4
Find the intersection
−22+4≤a≤22+4
2∣2a−4∣−8a2−4(2a−4)2>0,−22+4≤a≤22+4
Simplify
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Evaluate
2∣2a−4∣−8a2−4(2a−4)2
Calculate the absolute value
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Calculate
∣2a−4∣
Rewrite the expression
∣2(a−2)∣
Rewrite the expression
2∣a−2∣
2×2∣a−2∣−8a2−4(2a−4)2
Subtract the terms
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Simplify
8a2−4(2a−4)2
Expand the expression
8a2−16a2+64a−64
Subtract the terms
−8a2+64a−64
2×2∣a−2∣−−8a2+64a−64
Simplify the root
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Evaluate
−8a2+64a−64
Factor the expression
8(−a2+8a−8)
The root of a product is equal to the product of the roots of each factor
8×−a2+8a−8
Evaluate the root
22×−a2+8a−8
Calculate the product
2−2a2+16a−16
2×2∣a−2∣−2−2a2+16a−16
Multiply the terms
4∣a−2∣−2−2a2+16a−16
4∣a−2∣−2−2a2+16a−16>0
Separate the inequality into 2 possible cases
4(a−2)−2−2a2+16a−16>0,a−2≥04(−(a−2))−2−2a2+16a−16>0,a−2<0
Evaluate
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Evaluate
4(a−2)−2−2a2+16a−16>0
Simplify the expression
4a−8−2−2a2+16a−16>0
Move the expression to the right side
−2−2a2+16a−16>−4a+8
Divide both sides
−2a2+16a−16<−2(−a+2)
Separate the inequality into 2 possible cases
−2a2+16a−16<−2(−a+2),−2(−a+2)≥0−2a2+16a−16<−2(−a+2),−2(−a+2)<0
Solve the inequality
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Solve the inequality
−2a2+16a−16<−2(−a+2)
Square both sides of the inequality
−2a2+16a−16<(−2(−a+2))2
Calculate the sum or difference
−2a2+16a−16=(−2(−a+2))2
Calculate
−2a2+16a−16=4a2−16a+16
Move the expression to the left side
−2a2+16a−16−(4a2−16a+16)<0
Calculate the sum or difference
−6a2+32a−32<0
Move the constant to the right side
−6a2+32a<0−(−32)
Add the terms
−6a2+32a<32
Evaluate
a2−316a>−316
Add the same value to both sides
a2−316a+964>−316+964
Evaluate
a2−316a+964>916
Evaluate
(a−38)2>916
Take the 2-th root on both sides of the inequality
(a−38)2>916
Calculate
a−38>34
Separate the inequality into 2 possible cases
a−38>34a−38<−34
Calculate
a>4a−38<−34
Calculate
a>4a<34
Find the union
a∈(−∞,34)∪(4,+∞)
a∈(−∞,34)∪(4,+∞),−2(−a+2)≥0−2a2+16a−16<−2(−a+2),−2(−a+2)<0
Solve the inequality
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Evaluate
−2(−a+2)≥0
Change the signs on both sides of the inequality and flip the inequality sign
2(−a+2)≤0
Rewrite the expression
−a+2≤0
Move the constant to the right side
−a≤0−2
Removing 0 doesn't change the value,so remove it from the expression
−a≤−2
Change the signs on both sides of the inequality and flip the inequality sign
a≥2
a∈(−∞,34)∪(4,+∞),a≥2−2a2+16a−16<−2(−a+2),−2(−a+2)<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∈(−∞,34)∪(4,+∞),a≥2a∈∅,−2(−a+2)<0
Solve the inequality
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Evaluate
−2(−a+2)<0
Change the signs on both sides of the inequality and flip the inequality sign
2(−a+2)>0
Rewrite the expression
−a+2>0
Move the constant to the right side
−a>0−2
Removing 0 doesn't change the value,so remove it from the expression
−a>−2
Change the signs on both sides of the inequality and flip the inequality sign
a<2
a∈(−∞,34)∪(4,+∞),a≥2a∈∅,a<2
Find the intersection
a>4a∈∅,a<2
Find the intersection
a>4a∈∅
Find the union
a>4
a>4,a−2≥04(−(a−2))−2−2a2+16a−16>0,a−2<0
Evaluate
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Evaluate
a−2≥0
Move the constant to the right side
a≥0+2
Removing 0 doesn't change the value,so remove it from the expression
a≥2
a>4,a≥24(−(a−2))−2−2a2+16a−16>0,a−2<0
Evaluate
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Evaluate
4(−(a−2))−2−2a2+16a−16>0
Remove the parentheses
4(−a+2)−2−2a2+16a−16>0
Simplify the expression
−4a+8−2−2a2+16a−16>0
Move the expression to the right side
−2−2a2+16a−16>4a−8
Divide both sides
−2a2+16a−16<−2(a−2)
Separate the inequality into 2 possible cases
−2a2+16a−16<−2(a−2),−2(a−2)≥0−2a2+16a−16<−2(a−2),−2(a−2)<0
Solve the inequality
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Solve the inequality
−2a2+16a−16<−2(a−2)
Square both sides of the inequality
−2a2+16a−16<(−2(a−2))2
Calculate the sum or difference
−2a2+16a−16=(−2(a−2))2
Calculate
−2a2+16a−16=4a2−16a+16
Move the expression to the left side
−2a2+16a−16−(4a2−16a+16)<0
Calculate the sum or difference
−6a2+32a−32<0
Move the constant to the right side
−6a2+32a<0−(−32)
Add the terms
−6a2+32a<32
Evaluate
a2−316a>−316
Add the same value to both sides
a2−316a+964>−316+964
Evaluate
a2−316a+964>916
Evaluate
(a−38)2>916
Take the 2-th root on both sides of the inequality
(a−38)2>916
Calculate
a−38>34
Separate the inequality into 2 possible cases
a−38>34a−38<−34
Calculate
a>4a−38<−34
Calculate
a>4a<34
Find the union
a∈(−∞,34)∪(4,+∞)
a∈(−∞,34)∪(4,+∞),−2(a−2)≥0−2a2+16a−16<−2(a−2),−2(a−2)<0
Solve the inequality
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Evaluate
−2(a−2)≥0
Change the signs on both sides of the inequality and flip the inequality sign
2(a−2)≤0
Rewrite the expression
a−2≤0
Move the constant to the right side
a≤0+2
Removing 0 doesn't change the value,so remove it from the expression
a≤2
a∈(−∞,34)∪(4,+∞),a≤2−2a2+16a−16<−2(a−2),−2(a−2)<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∈(−∞,34)∪(4,+∞),a≤2a∈∅,−2(a−2)<0
Solve the inequality
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Evaluate
−2(a−2)<0
Change the signs on both sides of the inequality and flip the inequality sign
2(a−2)>0
Rewrite the expression
a−2>0
Move the constant to the right side
a>0+2
Removing 0 doesn't change the value,so remove it from the expression
a>2
a∈(−∞,34)∪(4,+∞),a≤2a∈∅,a>2
Find the intersection
a<34a∈∅,a>2
Find the intersection
a<34a∈∅
Find the union
a<34
a>4,a≥2a<34,a−2<0
Evaluate
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Evaluate
a−2<0
Move the constant to the right side
a<0+2
Removing 0 doesn't change the value,so remove it from the expression
a<2
a>4,a≥2a<34,a<2
Find the intersection
a>4a<34,a<2
Find the intersection
a>4a<34
Find the union
a∈(−∞,34)∪(4,+∞)
Check if the solution is in the defined range
a∈(−∞,34)∪(4,+∞),−22+4≤a≤22+4
Solution
a∈[−22+4,34)∪(4,22+4]
Show Solution
