Card 1 of 20
EasyStudy card 1
Prompt
What does \(x<a\) mean?
Equality is excluded.
Answer
Every value strictly less than \(a\).
Example
Use an open endpoint at \(a\) and shade left.
MathChapter 4: Linear Inequalities and Graphs
Flashcards
Solving Inequalities Flashcards
Twenty concise retrieval cards for inequality language, rules, number lines, and traps.
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Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What does ‘at most \(a\)’ mean?
The boundary is allowed.
Answer
\(x\le a\).
Example
At most \(12\) means \(x\le12\).
Card 3 of 20
EasyStudy card 3
Prompt
What does ‘at least \(a\)’ mean?
Think minimum.
Answer
\(x\ge a\).
Example
At least \(5\) means \(x\ge5\).
Card 4 of 20
EasyStudy card 4
Prompt
Which operators use an open number-line endpoint?
Strict inequalities exclude equality.
Answer
\(<\) and \(>\).
Example
\(x>2\) is open at \(2\).
Card 5 of 20
EasyStudy card 5
Prompt
Which operators use a closed number-line endpoint?
Equality is included.
Answer
\(\le\) and \(\ge\).
Example
\(x\le4\) is closed at \(4\).
Card 6 of 20
MediumStudy card 6
Prompt
When must an inequality sign reverse?
It is an order reflection.
Answer
When both sides are multiplied or divided by a negative number.
Example
From \(-2x<8\), obtain \(x>-4\).
Card 7 of 20
MediumStudy card 7
Prompt
Why does multiplying by a negative reverse order?
Compare \(2<5\).
Answer
It reflects values across zero, reversing their left-to-right order.
Example
After multiplying by \(-1\), \(-2>-5\).
Card 8 of 20
MediumStudy card 8
Prompt
Does adding a negative number reverse an inequality?
Only multiplication or division can trigger reversal.
Answer
No. Addition and subtraction preserve direction.
Example
From \(x+(-3)<7\), add \(3\) without reversing.
Card 9 of 20
MediumStudy card 9
Prompt
How do you solve a multi-step inequality?
Use equation-like steps.
Answer
Simplify, collect variables, collect constants, then isolate while monitoring the final divisor's sign.
Example
Reverse only if the isolating division is negative.
Card 10 of 20
MediumStudy card 10
Prompt
What is the transitive inequality property?
It chains order.
Answer
If \(a<b\) and \(b<c\), then \(a<c\).
Example
If \(p<4\) and \(4<9\), then \(p<9\).
Card 11 of 20
MediumStudy card 11
Prompt
Translate ‘does not exceed \(k\).’
Exceed means go above.
Answer
The quantity is \(\le k\).
Example
A load that does not exceed \(60\) satisfies \(w\le60\).
Card 12 of 20
MediumStudy card 12
Prompt
Translate ‘no less than \(k\).’
It cannot be below the boundary.
Answer
The quantity is \(\ge k\).
Example
No less than \(20\) means \(n\ge20\).
Card 13 of 20
MediumStudy card 13
Prompt
How is an inequality solution different from an equation solution?
Think solution set.
Answer
It usually describes an interval or ray of values, not one isolated value.
Example
\(x>3\) has infinitely many real solutions.
Card 14 of 20
MediumStudy card 14
Prompt
If integer \(n<8\), what is its greatest possible value?
Integers are discrete.
Answer
\(7\).
Example
There is no greatest real number below \(8\), but the greatest integer is \(7\).
Card 15 of 20
MediumStudy card 15
Prompt
What context check follows solving an inequality?
Units matter.
Answer
Apply stated domain restrictions such as whole, integer, or nonnegative values.
Example
A package count cannot be \(4.5\).
Card 16 of 20
MediumStudy card 16
Prompt
How should you check a solved inequality?
Use the original inequality.
Answer
Test one value inside and one value outside the proposed solution set.
Example
For \(x>-2\), test \(0\) and \(-3\).
Card 17 of 20
MediumStudy card 17
Prompt
What is a common trap with variable terms on both sides?
Track the remaining coefficient.
Answer
Creating a negative coefficient and forgetting the reversal during final division.
Example
Move terms to keep a positive coefficient when convenient.
Card 18 of 20
MediumStudy card 18
Prompt
How can fractions be handled efficiently?
A positive multiplier preserves direction.
Answer
Multiply every term by a positive common denominator.
Example
Multiply \(x/3+2>5\) by \(3\).
Card 19 of 20
MediumStudy card 19
Prompt
What does a right-pointing ray represent?
Right means larger.
Answer
Values greater than the endpoint, with inclusion determined by the endpoint marker.
Example
Closed at \(1\) and right means \(x\ge1\).
Card 20 of 20
MediumStudy card 20
Prompt
What does a left-pointing ray represent?
Left means smaller.
Answer
Values less than the endpoint, with inclusion determined by the endpoint marker.
Example
Open at \(-2\) and left means \(x<-2\).
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