Card 1 of 20
EasyStudy card 1
Prompt
What does AND mean in a compound inequality?
Look for overlap.
Answer
A value must satisfy both conditions: use the intersection.
Example
\(-2<x\) and \(x\le4\) becomes \(-2<x\le4\).
MathChapter 4: Linear Inequalities and Graphs
Flashcards
Compound and Absolute Value Inequalities Flashcards
Twenty retrieval cards for intersections, unions, distance forms, endpoints, and edge cases.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What does OR mean in a compound inequality?
Keep either valid part.
Answer
A value must satisfy at least one condition: use the union.
Example
\(x<-3\) or \(x\ge2\) gives two rays.
Card 3 of 20
EasyStudy card 3
Prompt
What is an intersection?
Logical AND.
Answer
The values shared by every solution set.
Example
The overlap of \(x>-1\) and \(x<5\) is \(-1<x<5\).
Card 4 of 20
EasyStudy card 4
Prompt
What is a union?
Logical OR.
Answer
All values belonging to at least one solution set.
Example
Combine separated rays without filling the gap.
Card 5 of 20
EasyStudy card 5
Prompt
How do you solve a chained inequality?
Keep the chain balanced.
Answer
Apply each operation to all three parts.
Example
From \(-3<2x+1\le9\), subtract \(1\) everywhere.
Card 6 of 20
MediumStudy card 6
Prompt
For \(a>0\), what is \(|E|<a\) equivalent to?
Small distance means inside.
Answer
\(-a<E<a\).
Example
\(|x|<4\) gives \(-4<x<4\).
Card 7 of 20
MediumStudy card 7
Prompt
For \(a>0\), what is \(|E|\le a\) equivalent to?
Include boundary distances.
Answer
\(-a\le E\le a\).
Example
\(|x|\le3\) gives \(-3\le x\le3\).
Card 8 of 20
MediumStudy card 8
Prompt
For \(a>0\), what is \(|E|>a\) equivalent to?
Large distance means outside.
Answer
\(E<-a\) or \(E>a\).
Example
\(|x|>2\) gives two open rays.
Card 9 of 20
MediumStudy card 9
Prompt
For \(a>0\), what is \(|E|\ge a\) equivalent to?
Outside includes boundaries.
Answer
\(E\le-a\) or \(E\ge a\).
Example
\(|x|\ge5\) includes \(-5\) and \(5\).
Card 10 of 20
MediumStudy card 10
Prompt
Why does absolute-value less-than use AND?
Distance is bounded on both sides.
Answer
The expression must lie between the two boundary values simultaneously.
Example
Distance from zero less than \(3\) stays between \(-3\) and \(3\).
Card 11 of 20
MediumStudy card 11
Prompt
Why does absolute-value greater-than use OR?
Two outside directions.
Answer
The expression may lie beyond either the negative boundary or the positive boundary.
Example
Distance greater than \(3\) means left of \(-3\) or right of \(3\).
Card 12 of 20
MediumStudy card 12
Prompt
What must happen before splitting an absolute-value inequality?
Remove outside constants first.
Answer
Isolate the absolute-value expression.
Example
From \(2|x|+1<9\), first obtain \(|x|<4\).
Card 13 of 20
MediumStudy card 13
Prompt
What is the solution of \(|E|<0\)?
Distance cannot be negative.
Answer
No solution.
Example
No real absolute value is strictly below zero.
Card 14 of 20
MediumStudy card 14
Prompt
What is the solution of \(|E|\ge0\)?
Absolute value is nonnegative.
Answer
All real inputs for which the expression is defined.
Example
\(|2x-1|\ge0\) is always true.
Card 15 of 20
MediumStudy card 15
Prompt
What is the solution of \(|E|\le0\)?
Only zero has distance zero.
Answer
Exactly the values making \(E=0\).
Example
\(|x+4|\le0\) gives \(x=-4\).
Card 16 of 20
MediumStudy card 16
Prompt
What endpoint mistake commonly occurs with compounds?
Carry the original equality status.
Answer
Changing a strict endpoint to closed or an inclusive endpoint to open.
Example
\(<\) remains open after adding a number.
Card 17 of 20
MediumStudy card 17
Prompt
How do integer restrictions change a compound solution?
List them explicitly.
Answer
Filter the real interval to the integers it contains.
Example
\(-1<x\le3\) gives integers \(0,1,2,3\).
Card 18 of 20
MediumStudy card 18
Prompt
Can an OR inequality simplify to one ray?
Take the union.
Answer
Yes, when one branch contains the other.
Example
\(x>2\) or \(x>5\) simplifies to \(x>2\).
Card 19 of 20
MediumStudy card 19
Prompt
Can an AND inequality have no solution?
Take the intersection.
Answer
Yes, when its required intervals do not overlap.
Example
\(x<1\) and \(x>4\) is impossible.
Card 20 of 20
MediumStudy card 20
Prompt
What is the main SAT check after solving two branches?
Inside uses intersection; outside uses union.
Answer
Confirm the connector and endpoint inclusion match the original condition.
Example
Do not write \(|x|>3\) as \(-3<x<3\).
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