Card 1 of 20
EasyStudy card 1
Prompt
What is a system of inequalities?
Every constraint applies.
Answer
Two or more inequalities involving the same variables.
Example
A feasible point must satisfy the complete set.
MathChapter 4: Linear Inequalities and Graphs
Flashcards
Graphing Systems of Inequalities Flashcards
Twenty retrieval cards for intersections, candidate points, regions, boundary inclusion, and modeling.
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Card 2 of 20
EasyStudy card 2
Prompt
What represents a system's solution set?
Logical AND.
Answer
The intersection of all individual solution regions.
Example
Keep only the shared overlap.
Card 3 of 20
EasyStudy card 3
Prompt
How do you test a candidate point?
One failure rejects it.
Answer
Substitute it into every inequality; every statement must be true.
Example
Make a true/false checklist.
Card 4 of 20
EasyStudy card 4
Prompt
Does satisfying one inequality make a point a system solution?
A system is not OR.
Answer
No. It must satisfy all inequalities.
Example
A point outside one half-plane is rejected.
Card 5 of 20
EasyStudy card 5
Prompt
What does an empty overlap mean?
The constraints are incompatible.
Answer
The system has no solution.
Example
\(y>4\) and \(y<1\) cannot both hold.
Card 6 of 20
MediumStudy card 6
Prompt
How are mixed boundary styles handled?
Do not give every line one style.
Answer
Choose solid or dashed separately from each inequality's operator.
Example
One system may contain both styles.
Card 7 of 20
MediumStudy card 7
Prompt
Can a dashed boundary segment belong to the solution?
Even if other constraints allow it.
Answer
No, because its strict inequality excludes equality.
Example
Points on \(y<x\)'s boundary are excluded.
Card 8 of 20
MediumStudy card 8
Prompt
Can a solid boundary segment belong to the solution?
Inclusion is necessary, not sufficient.
Answer
Yes, but only where it also satisfies every other inequality.
Example
Part of a solid line may lie outside another half-plane.
Card 9 of 20
MediumStudy card 9
Prompt
What is the reliable system graphing order?
Repeat before combining.
Answer
Graph and shade each inequality, then identify their common intersection.
Example
Verify a point from the overlap.
Card 10 of 20
MediumStudy card 10
Prompt
How do labeled-region questions work?
Do not rely only on visual darkness.
Answer
Test a representative point from each candidate region in every inequality.
Example
Choose points away from boundaries.
Card 11 of 20
MediumStudy card 11
Prompt
How can sign constraints identify a quadrant?
Quadrant IV has positive x and negative y.
Answer
Combine x and y restrictions with quadrant sign patterns.
Example
\(x>0\) and \(y<0\) force Quadrant IV.
Card 12 of 20
MediumStudy card 12
Prompt
What is the difference between union and intersection shading?
Systems use AND by default.
Answer
Union accepts at least one condition; a system requires the intersection satisfying all conditions.
Example
Do not keep every pastel area.
Card 13 of 20
MediumStudy card 13
Prompt
What does a vertical system boundary control?
Its equation is \(x=a\).
Answer
A restriction on x and therefore left/right position.
Example
\(x\ge2\) shades right of a solid vertical line.
Card 14 of 20
MediumStudy card 14
Prompt
What does a horizontal system boundary control?
Its equation is \(y=b\).
Answer
A restriction on y and therefore below/above position.
Example
\(y<4\) shades below a dashed horizontal line.
Card 15 of 20
MediumStudy card 15
Prompt
Why verify a graph intersection algebraically?
Use a simple interior point.
Answer
Visual overlap can be misread near boundaries, while substitution checks every constraint exactly.
Example
Do not infer inclusion from color alone.
Card 16 of 20
MediumStudy card 16
Prompt
Can a system contain three or more inequalities?
Each new constraint may shrink the region.
Answer
Yes. Continue intersecting until every constraint has been applied.
Example
A feasible polygon can have several boundary edges.
Card 17 of 20
MediumStudy card 17
Prompt
How do contextual systems arise?
All real-world constraints apply together.
Answer
Each resource, minimum, maximum, or relationship becomes a separate inequality.
Example
Also include nonnegative counts when stated or inherently required.
Card 18 of 20
MediumStudy card 18
Prompt
What is a common overlap-shading error?
Darker is not automatically correct.
Answer
Highlighting the union of individual regions instead of their shared intersection.
Example
Test a point in the proposed final region.
Card 19 of 20
MediumStudy card 19
Prompt
What if two parallel inequalities point away from each other?
Compare the required ranges.
Answer
Their intersection may be empty.
Example
\(y>5\) and \(y<2\) has no solution.
Card 20 of 20
MediumStudy card 20
Prompt
What is the fastest SAT check for a proposed system solution?
Direct testing avoids redrawing.
Answer
Substitute the ordered pair into each inequality and stop at the first false statement.
Example
If all are true, the point is in the intersection.
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