Algebra
Calculus
Trigonometry
Matrix
Question
Solve the inequality
x∈(−∞,−5]∪(−3,3)∪[5,+∞)
Evaluate
∣x∣−31≤21
Find the domain
More Steps
Evaluate
∣x∣−3=0
Rewrite the expression
∣x∣=3
Separate the inequality into 2 possible cases
{x=3x=−3
Find the intersection
x∈(−∞,−3)∪(−3,3)∪(3,+∞)
∣x∣−31≤21,x∈(−∞,−3)∪(−3,3)∪(3,+∞)
Move the expression to the left side
∣x∣−31−21≤0
Subtract the terms
More Steps
Evaluate
∣x∣−31−21
Reduce fractions to a common denominator
(∣x∣−3)×22−2(∣x∣−3)∣x∣−3
Use the commutative property to reorder the terms
2(∣x∣−3)2−2(∣x∣−3)∣x∣−3
Write all numerators above the common denominator
2(∣x∣−3)2−(∣x∣−3)
Subtract the terms
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Evaluate
2−(∣x∣−3)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
2−∣x∣+3
Add the numbers
5−∣x∣
2(∣x∣−3)5−∣x∣
Multiply the terms
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Evaluate
2(∣x∣−3)
Use the the distributive property to expand the expression
2∣x∣+2(−3)
Multiply the numbers
2∣x∣−6
2∣x∣−65−∣x∣
2∣x∣−65−∣x∣≤0
Separate the inequality into 2 possible cases
{5−∣x∣≥02∣x∣−6<0{5−∣x∣≤02∣x∣−6>0
Solve the inequality
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Evaluate
5−∣x∣≥0
Rearrange the terms
−∣x∣≥−5
Calculate
∣x∣≤5
Separate the inequality into 2 possible cases
{x≤5x≥−5
Find the intersection
−5≤x≤5
{−5≤x≤52∣x∣−6<0{5−∣x∣≤02∣x∣−6>0
Solve the inequality
More Steps
Evaluate
2∣x∣−6<0
Rewrite the expression
2∣x∣<6
Divide both sides
22∣x∣<26
Divide the numbers
∣x∣<26
Divide the numbers
More Steps
Evaluate
26
Reduce the numbers
13
Calculate
3
∣x∣<3
Separate the inequality into 2 possible cases
{x<3x>−3
Find the intersection
−3<x<3
{−5≤x≤5−3<x<3{5−∣x∣≤02∣x∣−6>0
Solve the inequality
More Steps
Evaluate
5−∣x∣≤0
Rearrange the terms
−∣x∣≤−5
Calculate
∣x∣≥5
Separate the inequality into 2 possible cases
x≥5x≤−5
Find the union
x∈(−∞,−5]∪[5,+∞)
{−5≤x≤5−3<x<3{x∈(−∞,−5]∪[5,+∞)2∣x∣−6>0
Solve the inequality
More Steps
Evaluate
2∣x∣−6>0
Rewrite the expression
2∣x∣>6
Divide both sides
22∣x∣>26
Divide the numbers
∣x∣>26
Divide the numbers
More Steps
Evaluate
26
Reduce the numbers
13
Calculate
3
∣x∣>3
Separate the inequality into 2 possible cases
x>3x<−3
Find the union
x∈(−∞,−3)∪(3,+∞)
{−5≤x≤5−3<x<3{x∈(−∞,−5]∪[5,+∞)x∈(−∞,−3)∪(3,+∞)
Find the intersection
−3<x<3{x∈(−∞,−5]∪[5,+∞)x∈(−∞,−3)∪(3,+∞)
Find the intersection
−3<x<3x∈(−∞,−5]∪[5,+∞)
Find the union
x∈(−∞,−5]∪(−3,3)∪[5,+∞)
Check if the solution is in the defined range
x∈(−∞,−5]∪(−3,3)∪[5,+∞),x∈(−∞,−3)∪(−3,3)∪(3,+∞)
Solution
x∈(−∞,−5]∪(−3,3)∪[5,+∞)
Show Solution
