Practice
Solving Word Problems Using Inequalities Practice
Fifty original questions covering constraint language, budgets, number lines, capacities, pricing comparisons, integer interpretation, and strict boundaries.
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Question 1
Explanation
The wording determines direction and whether equality is allowed. The correct constraint is \(b\le64\).
Question 2
Explanation
The wording determines direction and whether equality is allowed. The correct constraint is \(h\ge18\).
Question 3
Explanation
The wording determines direction and whether equality is allowed. The correct constraint is \(L<40\).
Question 4
Explanation
The wording determines direction and whether equality is allowed. The correct constraint is \(s>12\).
Question 5
Explanation
The wording determines direction and whether equality is allowed. The correct constraint is \(m\le250\).
Question 6
Explanation
Let the variable be the requested whole count. Translate the budget as \(25+6u\le121\); solving and taking the greatest feasible whole value gives \(16\).
Question 7
Explanation
Let the variable be the requested whole count. Translate the budget as \(48+8.5h\le150\); solving and taking the greatest feasible whole value gives \(12\).
Question 8
Explanation
Let the variable be the requested whole count. Translate the budget as \(75+12p\le255\); solving and taking the greatest feasible whole value gives \(15\).
Question 9
Explanation
Let the variable be the requested whole count. Translate the budget as \(32+4.5m\le95\); solving and taking the greatest feasible whole value gives \(14\).
Question 10
Explanation
Let the variable be the requested whole count. Translate the budget as \(120+15p\le420\); solving and taking the greatest feasible whole value gives \(20\).
Question 11
Explanation
The statement translates to \(x>=30\). Endpoint inclusion and direction produce the displayed solution set.
Question 12
Explanation
The statement translates to \(x<=12\). Endpoint inclusion and direction produce the displayed solution set.
Question 13
Explanation
The statement translates to \(x>8\). Endpoint inclusion and direction produce the displayed solution set.
Question 14
Explanation
The statement translates to \(x<50\). Endpoint inclusion and direction produce the displayed solution set.
Question 15
Explanation
The statement translates to \(x>=20\). Endpoint inclusion and direction produce the displayed solution set.
Question 16
Explanation
Translate the contextual limit as \(14n-90\ge400\). Solve, then select the required minimum or maximum feasible integer, \(35\).
Question 17
Explanation
Translate the contextual limit as \(18+6c\le150\). Solve, then select the required minimum or maximum feasible integer, \(22\).
Question 18
Explanation
Translate the contextual limit as \(22h\ge550\). Solve, then select the required minimum or maximum feasible integer, \(25\).
Question 19
Explanation
Translate the contextual limit as \(240-9d\ge60\). Solve, then select the required minimum or maximum feasible integer, \(20\).
Question 20
Explanation
Translate the contextual limit as \(45+7m<200\). Solve, then select the required minimum or maximum feasible integer, \(22\).
Question 21
Explanation
Let \(u\) be whole usage and compare in the requested direction: \(54+3u<30+5u\). The strict crossover makes \(13\) the first feasible whole usage.
Question 22
Explanation
Let \(u\) be whole usage and compare in the requested direction: \(80+2u<20+5u\). The strict crossover makes \(21\) the first feasible whole usage.
Question 23
Explanation
Let \(u\) be whole usage and compare in the requested direction: \(51+6u<15+9u\). The strict crossover makes \(13\) the first feasible whole usage.
Question 24
Explanation
Let \(u\) be whole usage and compare in the requested direction: \(100+4u<40+7u\). The strict crossover makes \(21\) the first feasible whole usage.
Question 25
Explanation
Let \(u\) be whole usage and compare in the requested direction: \(65+3u<25+8u\). The strict crossover makes \(9\) the first feasible whole usage.
Question 26
Explanation
Translate the contextual boundary as \(160+28e\ge356\). Solving and applying whole-number feasibility gives \(7\).
Question 27
Explanation
Translate the contextual boundary as \(500-32h>180\), so \(h<10\). Solving and applying whole-number feasibility gives \(9\).
Question 28
Explanation
Translate the contextual boundary as \(35+7g\le420\). Solving and applying whole-number feasibility gives \(55\).
Question 29
Explanation
Translate the contextual boundary as \(48+16d\ge160\). Solving and applying whole-number feasibility gives \(7\).
Question 30
Explanation
Translate the contextual boundary as \(12+5n<72\), so \(n<12\). Solving and applying whole-number feasibility gives \(11\).
Question 31
Explanation
Move up to the first whole number greater than the boundary. This is constraint-aware integer selection, not ordinary rounding, so the answer is \(417\).
Question 32
Explanation
Move down to the greatest whole number below the boundary. This is constraint-aware integer selection, not ordinary rounding, so the answer is \(11\).
Question 33
Explanation
The first whole number at least \(24.2\) is \(25\). This is constraint-aware integer selection, not ordinary rounding, so the answer is \(25\).
Question 34
Explanation
The greatest whole number not exceeding \(73.9\) is \(73\). This is constraint-aware integer selection, not ordinary rounding, so the answer is \(73\).
Question 35
Explanation
The boundary \(9\) is excluded, so the next whole number is \(10\). This is constraint-aware integer selection, not ordinary rounding, so the answer is \(10\).
Question 36
Explanation
The lines meet at usage \(10\). Equality is not cheaper, so the least whole usage beyond the crossover is \(11\).
Question 37
Explanation
The lines meet at usage \(10\). Equality is not cheaper, so the least whole usage beyond the crossover is \(11\).
Question 38
Explanation
The lines meet at usage \(10\). Equality is not cheaper, so the least whole usage beyond the crossover is \(11\).
Question 39
Explanation
The lines meet at usage \(20\). Equality is not cheaper, so the least whole usage beyond the crossover is \(21\).
Question 40
Explanation
The lines meet at usage \(15\). Equality is not cheaper, so the least whole usage beyond the crossover is \(16\).
Question 41
Explanation
Boundary inclusion follows the exact phrase. Yes; ‘at most’ includes the boundary.
Question 42
Explanation
Boundary inclusion follows the exact phrase. No; ‘fewer than’ is strict.
Question 43
Explanation
Boundary inclusion follows the exact phrase. Yes; a minimum means \(m\ge15\).
Question 44
Explanation
Boundary inclusion follows the exact phrase. No; ‘more than’ excludes the boundary.
Question 45
Explanation
Boundary inclusion follows the exact phrase. Yes; ‘no less than’ means \(u\ge42\).
Question 46
Explanation
Solve the exact inequality: \(96-6h\ge30\), so \(-6h\ge-66\) and \(h\le11\). Apply strictness and the whole-number domain to obtain \(11\).
Question 47
Explanation
Solve the exact inequality: \(17n-230\ge450\), so \(17n\ge680\). Apply strictness and the whole-number domain to obtain \(40\).
Question 48
Explanation
Solve the exact inequality: \(28+0.12m\le70\), so \(m\le350\). Apply strictness and the whole-number domain to obtain \(350\).
Question 49
Explanation
Solve the exact inequality: \(75+0.25m<45+0.40m\), so \(m>200\). Apply strictness and the whole-number domain to obtain \(201\).
Question 50
Explanation
Solve the exact inequality: \(95+13.5p<300\), so \(p<15.185\ldots\). Apply strictness and the whole-number domain to obtain \(15\).
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Questions to review
No mistakes this time. Excellent work.
- Question 1Contextual inequality languageEasy
- Question 2Contextual inequality languageEasy
- Question 3Contextual inequality languageEasy
- Question 4Contextual inequality languageEasy
- Question 5Contextual inequality languageEasy
- Question 6Budget constraintsEasy
- Question 7Budget constraintsEasy
- Question 8Budget constraintsEasy
- Question 9Budget constraintsEasy
- Question 10Budget constraintsEasy
- Question 11Contextual number-line interpretationEasy
- Question 12Contextual number-line interpretationEasy
- Question 13Contextual number-line interpretationEasy
- Question 14Contextual number-line interpretationEasy
- Question 15Contextual number-line interpretationEasy
- Question 16Capacity and minimum constraintsMedium
- Question 17Capacity and minimum constraintsMedium
- Question 18Capacity and minimum constraintsMedium
- Question 19Capacity and minimum constraintsMedium
- Question 20Capacity and minimum constraintsMedium
- Question 21Pricing comparison inequalitiesMedium
- Question 22Pricing comparison inequalitiesMedium
- Question 23Pricing comparison inequalitiesMedium
- Question 24Pricing comparison inequalitiesMedium
- Question 25Pricing comparison inequalitiesMedium
- Question 26Solving contextual inequalitiesMedium
- Question 27Solving contextual inequalitiesMedium
- Question 28Solving contextual inequalitiesMedium
- Question 29Solving contextual inequalitiesMedium
- Question 30Solving contextual inequalitiesMedium
- Question 31Integer constraint interpretationMedium
- Question 32Integer constraint interpretationMedium
- Question 33Integer constraint interpretationMedium
- Question 34Integer constraint interpretationMedium
- Question 35Integer constraint interpretationMedium
- Question 36Graphing pricing comparisonsMedium
- Question 37Graphing pricing comparisonsMedium
- Question 38Graphing pricing comparisonsMedium
- Question 39Graphing pricing comparisonsMedium
- Question 40Graphing pricing comparisonsMedium
- Question 41Strict versus inclusive constraintsHard
- Question 42Strict versus inclusive constraintsHard
- Question 43Strict versus inclusive constraintsHard
- Question 44Strict versus inclusive constraintsHard
- Question 45Strict versus inclusive constraintsHard
- Question 46Challenging inequality applicationsHard
- Question 47Challenging inequality applicationsHard
- Question 48Challenging inequality applicationsHard
- Question 49Challenging inequality applicationsHard
- Question 50Challenging inequality applicationsHard