Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,−4]∪[−1,1]∪[4,+∞)
Evaluate
x2−43x≤1
Find the domain
More Steps
Evaluate
x2−4=0
Move the constant to the right side
x2=4
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±4
Simplify the expression
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Evaluate
4
Write the number in exponential form with the base of 2
22
Reduce the index of the radical and exponent with 2
2
x=±2
Separate the inequality into 2 possible cases
{x=2x=−2
Find the intersection
x∈(−∞,−2)∪(−2,2)∪(2,+∞)
x2−43x≤1,x∈(−∞,−2)∪(−2,2)∪(2,+∞)
Simplify
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Evaluate
x2−43x
Rewrite the expression
3×x2−4x
Rewrite the expression
3x2−4x
3x2−4x≤1
Calculate
∣x2−4∣3∣x∣−x2−4≤0
Separate the inequality into 2 possible cases
{3∣x∣−x2−4≥0x2−4<0{3∣x∣−x2−4≤0x2−4>0
Solve the inequality
More Steps
Evaluate
3∣x∣−x2−4≥0
Separate the inequality into 4 possible cases
3x−(x2−4)≥0,x≥0,x2−4≥03x−(−(x2−4))≥0,x≥0,x2−4<03(−x)−(x2−4)≥0,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3x−(x2−4)≥0
Remove the parentheses
3x−x2+4≥0
Move the constant to the right side
3x−x2≥0−4
Add the terms
3x−x2≥−4
Evaluate
x2−3x≤4
Add the same value to both sides
x2−3x+49≤4+49
Evaluate
x2−3x+49≤425
Evaluate
(x−23)2≤425
Take the 2-th root on both sides of the inequality
(x−23)2≤425
Calculate
x−23≤25
Separate the inequality into 2 possible cases
{x−23≤25x−23≥−25
Calculate
{x≤4x−23≥−25
Calculate
{x≤4x≥−1
Find the intersection
−1≤x≤4
−1≤x≤4,x≥0,x2−4≥03x−(−(x2−4))≥0,x≥0,x2−4<03(−x)−(x2−4)≥0,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4≥0
Move the constant to the right side
x2≥4
Take the 2-th root on both sides of the inequality
x2≥4
Calculate
∣x∣≥2
Separate the inequality into 2 possible cases
x≥2x≤−2
Find the union
x∈(−∞,−2]∪[2,+∞)
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)3x−(−(x2−4))≥0,x≥0,x2−4<03(−x)−(x2−4)≥0,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3x−(−(x2−4))≥0
Remove the parentheses
3x+x2−4≥0
Move the constant to the right side
3x+x2≥0−(−4)
Add the terms
3x+x2≥4
Evaluate
x2+3x≥4
Add the same value to both sides
x2+3x+49≥4+49
Evaluate
x2+3x+49≥425
Evaluate
(x+23)2≥425
Take the 2-th root on both sides of the inequality
(x+23)2≥425
Calculate
x+23≥25
Separate the inequality into 2 possible cases
x+23≥25x+23≤−25
Calculate
x≥1x+23≤−25
Calculate
x≥1x≤−4
Find the union
x∈(−∞,−4]∪[1,+∞)
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,x2−4<03(−x)−(x2−4)≥0,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4<0
Move the constant to the right side
x2<4
Take the 2-th root on both sides of the inequality
x2<4
Calculate
∣x∣<2
Separate the inequality into 2 possible cases
{x<2x>−2
Find the intersection
−2<x<2
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<23(−x)−(x2−4)≥0,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3(−x)−(x2−4)≥0
Remove the parentheses
3(−x)−x2+4≥0
Simplify the expression
−3x−x2+4≥0
Move the constant to the right side
−3x−x2≥0−4
Add the terms
−3x−x2≥−4
Evaluate
x2+3x≤4
Add the same value to both sides
x2+3x+49≤4+49
Evaluate
x2+3x+49≤425
Evaluate
(x+23)2≤425
Take the 2-th root on both sides of the inequality
(x+23)2≤425
Calculate
x+23≤25
Separate the inequality into 2 possible cases
{x+23≤25x+23≥−25
Calculate
{x≤1x+23≥−25
Calculate
{x≤1x≥−4
Find the intersection
−4≤x≤1
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<2−4≤x≤1,x<0,x2−4≥03(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4≥0
Move the constant to the right side
x2≥4
Take the 2-th root on both sides of the inequality
x2≥4
Calculate
∣x∣≥2
Separate the inequality into 2 possible cases
x≥2x≤−2
Find the union
x∈(−∞,−2]∪[2,+∞)
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<2−4≤x≤1,x<0,x∈(−∞,−2]∪[2,+∞)3(−x)−(−(x2−4))≥0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3(−x)−(−(x2−4))≥0
Remove the parentheses
3(−x)+x2−4≥0
Simplify the expression
−3x+x2−4≥0
Move the constant to the right side
−3x+x2≥0−(−4)
Add the terms
−3x+x2≥4
Evaluate
x2−3x≥4
Add the same value to both sides
x2−3x+49≥4+49
Evaluate
x2−3x+49≥425
Evaluate
(x−23)2≥425
Take the 2-th root on both sides of the inequality
(x−23)2≥425
Calculate
x−23≥25
Separate the inequality into 2 possible cases
x−23≥25x−23≤−25
Calculate
x≥4x−23≤−25
Calculate
x≥4x≤−1
Find the union
x∈(−∞,−1]∪[4,+∞)
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<2−4≤x≤1,x<0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−1]∪[4,+∞),x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4<0
Move the constant to the right side
x2<4
Take the 2-th root on both sides of the inequality
x2<4
Calculate
∣x∣<2
Separate the inequality into 2 possible cases
{x<2x>−2
Find the intersection
−2<x<2
−1≤x≤4,x≥0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<2−4≤x≤1,x<0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−1]∪[4,+∞),x<0,−2<x<2
Find the intersection
2≤x≤4x∈(−∞,−4]∪[1,+∞),x≥0,−2<x<2−4≤x≤1,x<0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−1]∪[4,+∞),x<0,−2<x<2
Find the intersection
2≤x≤41≤x<2−4≤x≤1,x<0,x∈(−∞,−2]∪[2,+∞)x∈(−∞,−1]∪[4,+∞),x<0,−2<x<2
Find the intersection
2≤x≤41≤x<2−4≤x≤−2x∈(−∞,−1]∪[4,+∞),x<0,−2<x<2
Find the intersection
2≤x≤41≤x<2−4≤x≤−2−2<x≤−1
Find the union
x∈[−4,−1]∪[1,4]
{x∈[−4,−1]∪[1,4]x2−4<0{3∣x∣−x2−4≤0x2−4>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is false for any value of x
{x∈[−4,−1]∪[1,4]x∈/R{3∣x∣−x2−4≤0x2−4>0
Solve the inequality
More Steps
Evaluate
3∣x∣−x2−4≤0
Separate the inequality into 4 possible cases
3x−(x2−4)≤0,x≥0,x2−4≥03x−(−(x2−4))≤0,x≥0,x2−4<03(−x)−(x2−4)≤0,x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3x−(x2−4)≤0
Remove the parentheses
3x−x2+4≤0
Move the constant to the right side
3x−x2≤0−4
Add the terms
3x−x2≤−4
Evaluate
x2−3x≥4
Add the same value to both sides
x2−3x+49≥4+49
Evaluate
x2−3x+49≥425
Evaluate
(x−23)2≥425
Take the 2-th root on both sides of the inequality
(x−23)2≥425
Calculate
x−23≥25
Separate the inequality into 2 possible cases
x−23≥25x−23≤−25
Calculate
x≥4x−23≤−25
Calculate
x≥4x≤−1
Find the union
x∈(−∞,−1]∪[4,+∞)
x∈(−∞,−1]∪[4,+∞),x≥0,x2−4≥03x−(−(x2−4))≤0,x≥0,x2−4<03(−x)−(x2−4)≤0,x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4≥0
Move the constant to the right side
x2≥4
Take the 2-th root on both sides of the inequality
x2≥4
Calculate
∣x∣≥2
Separate the inequality into 2 possible cases
x≥2x≤−2
Find the union
x∈(−∞,−2]∪[2,+∞)
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)3x−(−(x2−4))≤0,x≥0,x2−4<03(−x)−(x2−4)≤0,x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3x−(−(x2−4))≤0
Remove the parentheses
3x+x2−4≤0
Move the constant to the right side
3x+x2≤0−(−4)
Add the terms
3x+x2≤4
Evaluate
x2+3x≤4
Add the same value to both sides
x2+3x+49≤4+49
Evaluate
x2+3x+49≤425
Evaluate
(x+23)2≤425
Take the 2-th root on both sides of the inequality
(x+23)2≤425
Calculate
x+23≤25
Separate the inequality into 2 possible cases
{x+23≤25x+23≥−25
Calculate
{x≤1x+23≥−25
Calculate
{x≤1x≥−4
Find the intersection
−4≤x≤1
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,x2−4<03(−x)−(x2−4)≤0,x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4<0
Move the constant to the right side
x2<4
Take the 2-th root on both sides of the inequality
x2<4
Calculate
∣x∣<2
Separate the inequality into 2 possible cases
{x<2x>−2
Find the intersection
−2<x<2
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,−2<x<23(−x)−(x2−4)≤0,x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3(−x)−(x2−4)≤0
Remove the parentheses
3(−x)−x2+4≤0
Simplify the expression
−3x−x2+4≤0
Move the constant to the right side
−3x−x2≤0−4
Add the terms
−3x−x2≤−4
Evaluate
x2+3x≥4
Add the same value to both sides
x2+3x+49≥4+49
Evaluate
x2+3x+49≥425
Evaluate
(x+23)2≥425
Take the 2-th root on both sides of the inequality
(x+23)2≥425
Calculate
x+23≥25
Separate the inequality into 2 possible cases
x+23≥25x+23≤−25
Calculate
x≥1x+23≤−25
Calculate
x≥1x≤−4
Find the union
x∈(−∞,−4]∪[1,+∞)
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,−2<x<2x∈(−∞,−4]∪[1,+∞),x<0,x2−4≥03(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4≥0
Move the constant to the right side
x2≥4
Take the 2-th root on both sides of the inequality
x2≥4
Calculate
∣x∣≥2
Separate the inequality into 2 possible cases
x≥2x≤−2
Find the union
x∈(−∞,−2]∪[2,+∞)
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,−2<x<2x∈(−∞,−4]∪[1,+∞),x<0,x∈(−∞,−2]∪[2,+∞)3(−x)−(−(x2−4))≤0,x<0,x2−4<0
Evaluate
More Steps
Evaluate
3(−x)−(−(x2−4))≤0
Remove the parentheses
3(−x)+x2−4≤0
Simplify the expression
−3x+x2−4≤0
Move the constant to the right side
−3x+x2≤0−(−4)
Add the terms
−3x+x2≤4
Evaluate
x2−3x≤4
Add the same value to both sides
x2−3x+49≤4+49
Evaluate
x2−3x+49≤425
Evaluate
(x−23)2≤425
Take the 2-th root on both sides of the inequality
(x−23)2≤425
Calculate
x−23≤25
Separate the inequality into 2 possible cases
{x−23≤25x−23≥−25
Calculate
{x≤4x−23≥−25
Calculate
{x≤4x≥−1
Find the intersection
−1≤x≤4
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,−2<x<2x∈(−∞,−4]∪[1,+∞),x<0,x∈(−∞,−2]∪[2,+∞)−1≤x≤4,x<0,x2−4<0
Evaluate
More Steps
Evaluate
x2−4<0
Move the constant to the right side
x2<4
Take the 2-th root on both sides of the inequality
x2<4
Calculate
∣x∣<2
Separate the inequality into 2 possible cases
{x<2x>−2
Find the intersection
−2<x<2
x∈(−∞,−1]∪[4,+∞),x≥0,x∈(−∞,−2]∪[2,+∞)−4≤x≤1,x≥0,−2<x<2x∈(−∞,−4]∪[1,+∞),x<0,x∈(−∞,−2]∪[2,+∞)−1≤x≤4,x<0,−2<x<2
Find the intersection
x≥4−4≤x≤1,x≥0,−2<x<2x∈(−∞,−4]∪[1,+∞),x<0,x∈(−∞,−2]∪[2,+∞)−1≤x≤4,x<0,−2<x<2
Find the intersection
x≥40≤x≤1x∈(−∞,−4]∪[1,+∞),x<0,x∈(−∞,−2]∪[2,+∞)−1≤x≤4,x<0,−2<x<2
Find the intersection
x≥40≤x≤1x≤−4−1≤x≤4,x<0,−2<x<2
Find the intersection
x≥40≤x≤1x≤−4−1≤x<0
Find the union
x∈(−∞,−4]∪[−1,1]∪[4,+∞)
{x∈[−4,−1]∪[1,4]x∈/R{x∈(−∞,−4]∪[−1,1]∪[4,+∞)x2−4>0
Solve the inequality
More Steps
Evaluate
x2−4>0
Since the left-hand side is always positive or 0,and the right-hand side is always 0,the statement is true for any value of x,except when x2−4=0
x2−4=0
Rewrite the expression
x2−4=0
Move the constant to the right-hand side and change its sign
x2=0+4
Removing 0 doesn't change the value,so remove it from the expression
x2=4
Take the root of both sides of the equation and remember to use both positive and negative roots
x=±4
Simplify the expression
More Steps
Evaluate
4
Write the number in exponential form with the base of 2
22
Reduce the index of the radical and exponent with 2
2
x=±2
Separate the equation into 2 possible cases
x=2x=−2
Exclude the impossible values of x
x∈(−∞,−2)∪(−2,2)∪(2,+∞)
{x∈[−4,−1]∪[1,4]x∈/R{x∈(−∞,−4]∪[−1,1]∪[4,+∞)x∈(−∞,−2)∪(−2,2)∪(2,+∞)
Find the intersection
x∈/R{x∈(−∞,−4]∪[−1,1]∪[4,+∞)x∈(−∞,−2)∪(−2,2)∪(2,+∞)
Find the intersection
x∈/Rx∈(−∞,−4]∪[−1,1]∪[4,+∞)
Find the union
x∈(−∞,−4]∪[−1,1]∪[4,+∞)
Check if the solution is in the defined range
x∈(−∞,−4]∪[−1,1]∪[4,+∞),x∈(−∞,−2)∪(−2,2)∪(2,+∞)
Solution
x∈(−∞,−4]∪[−1,1]∪[4,+∞)
Show Solution
