Write and Represent Simple Inequalities - 6th Grade Math

The inequality $x < -2$ has infinitely many solutions. Examine the number line representation below. Which statement about the solutions is correct?
Answer: All numbers to the left of -2, not including -2, are solutions to the inequality The inequality $x < -2$ means $x$ is strictly less than -2, so -2 itself is not included in the solution set. The solutions include all real numbers to the left of -2 on the number line, such as -3, -2.5, -10, etc. Choice A incorrectly includes -2. Choice C incorrectly limits solutions to only integers. Choice D describes the wrong direction (numbers greater than -2).
Let $x$ be any number. Which statement correctly describes the number of solutions to the inequality $x>10$ ?
Answer: It has infinitely many solutions (all numbers greater than 10). This question tests understanding that inequalities like x > 10 have infinitely many solutions, contrasting with equations, and relating to real-world infinite possibilities. Constraint words aren't direct here, but recall 'more than' → > excludes boundary, and solutions are infinite (x > 10: 10.1,11,... forever). Example: x > 10 has infinite solutions like 11,12,... and all reals in between, graph: ○ at 10 shaded right. Correct statement is it has infinitely many solutions (all >10), matching choice D. Error: thinking it's like equation with one solution, or finite like 10 solutions, or none. For such inequalities: recognize infinite set; graphing reinforces with shading to infinity. Context: any number greater satisfies; mistakes: confusing with equations, not recognizing infinity.
A club requires members to sell at least 10 tickets for a fundraiser. Let $x$ be the number of tickets sold. Which statement best describes the solutions to the inequality?
Answer: There are infinitely many solutions: any number $x$ with $x\ge 10$ . This question tests writing inequalities like x > c or x < c from real-world constraints, understanding that they have infinitely many solutions, and representing them on number lines with proper boundary markers (● for included, ○ for excluded) and shading. Constraint words translate to symbols: 'at least' means ≥ (includes boundary, like x ≥ 10 allows 10 or more), 'at most' means ≤ (includes), 'more than' means > (excludes), 'less than' means < (excludes); inequalities have infinite solutions (x ≥ 10 includes 10, 11, ... forever) unlike equations with one solution, and on a number line, use closed ● for ≥/≤, open ○ for >/<, shading right for greater and left for lesser. For example, 'at least 10 tickets' gives x ≥ 10 with infinite solutions, not finite like up to 20. The correct statement is there are infinitely many solutions: any x ≥ 10. A common error is claiming finite solutions or exactly one like an equation. Context: infinite possible ticket counts ≥ 10. Mistakes: not recognizing infinity, treating as equation.
A teacher says, "You may check out at most 6 books at a time." Let $b$ be the number of books checked out. Which value is not a solution to the inequality?
Answer: $b=7$ This question tests writing inequalities like b > c or b < c from real-world constraints, understanding that they have infinitely many solutions, and representing them on number lines with proper boundary markers (● for included, ○ for excluded) and shading. Constraint words translate to symbols: 'at least' means ≥ (includes), 'at most' means ≤ (includes boundary, like b ≤ 6 allows 6 or less), 'more than' means > (excludes), 'less than' means < (excludes); inequalities have infinite solutions unlike equations, and on a number line, use closed ● for ≥/≤, open ○ for >/<, shading right for greater and left for lesser. For example, 'at most 6' gives b ≤ 6 with solutions 0,1,2,...,6 infinitely many if reals, but here integers. The value not a solution is b=7, as it exceeds ≤6. A common error is thinking 6 is excluded like with <. Infinite solutions: all ≤6; context 'at most 6 books' allows 0 to 6. Mistakes: wrong symbol, confusing with more than.
Jordan is buying snacks and wants to spend at most $50. Let $x$ be the total cost (in dollars). Which inequality represents this situation?
Answer: $x\le 50$ This question tests writing inequalities like x > c or x < c from real-world constraints, understanding that they have infinitely many solutions, and representing them on number lines with proper boundary markers (● for included, ○ for excluded) and shading. Constraint words translate to symbols: 'at least' means ≥ (includes boundary, like x ≥ 10 allows 10 or more), 'at most' means ≤ (includes, like x ≤ 50 allows 50 or less), 'more than' means > (excludes), 'less than' means < (excludes); inequalities have infinite solutions unlike equations, and on a number line, use closed ● for ≥/≤, open ○ for >/<, shading right for greater and left for lesser. For example, 'score more than 75' gives x > 75 with solutions like 76, 77, ... infinitely many, graphed as ○ at 75 shaded right. The correct inequality is x ≤ 50, as 'at most 50' includes up to 50. A common error is using < instead of ≤, excluding 50, or reversing to ≥ thinking of minimum. To write it: (1) identify 'at most' as ≤, (2) translate to x ≤ 50. Infinite solutions: all reals ≤ 50; mistakes include confusing at most with more than or wrong symbol.
An elevator has a weight limit. The sign reads "Maximum capacity: 2000 pounds." If the elevator currently has 1650 pounds of weight, how much additional weight $w$ can be added?
Answer: $w \leq 350$ because the total weight cannot exceed the maximum capacity Current weight is 1650 pounds, maximum is 2000 pounds, so additional weight plus current weight cannot exceed 2000: $1650 + w ≤ 2000$ , which gives $w ≤ 350$ . Choice A uses strict inequality, but exactly 350 pounds additional would reach exactly the maximum (allowed). Choices C and D suggest more than 350 pounds can be added, which would exceed the limit.
To earn a badge in a video game, you must score more than 75 points. Let $s$ be your score. Which inequality represents all possible scores that earn the badge?
Answer: $s>75$ This question tests writing inequalities like s > c from real-world constraints, recognizing infinitely many solutions, and the difference in symbols for exclusive vs inclusive boundaries. Constraint words to symbols: 'more than' → > (excludes boundary: s > 75 means above 75, not 75), 'at least' → ≥ (includes), 'less than' → <, 'at most' → ≤; infinite solutions as s > 75 includes 75.1, 76, ..., forever. For example, 'score more than 75' → s > 75, solutions: 76, 77, ... infinite integers or reals, graph: open ○ at 75 shaded right (○======>). The correct inequality is s > 75, matching choice C. Errors include using ≥ instead of >, including 75 wrongly, or reversing to <75, or saying finite solutions like only up to 100. Writing: (1) identify 'more than' as exclusive greater, (2) translate to >, (3) write s > 75; graphing would be open ○ at 75, shade right. Infinite solutions: uncountably many reals >75; context: scores above 75 qualify; mistakes: symbol mix-up (> vs ≥), not recognizing infinity.
The inequality $y > -1$ is represented on the coordinate plane shown. A student claims that the point $(-2, 0)$ is a solution because "0 is greater than -1." Is the student correct?
Answer: Yes, because the y-coordinate 0 satisfies the inequality $y > -1$ regardless of x-coordinate The inequality $y > -1$ only constrains the y-coordinate, not the x-coordinate. Since the y-coordinate is 0 and $0 > -1$ is true, the point $(-2, 0)$ is indeed a solution. The x-coordinate of -2 is irrelevant to this inequality. Choice B is wrong about shaded regions. Choice C incorrectly relates x-coordinates to y-inequalities. Choice D incorrectly requires both coordinates to be positive.
A movie is rated for ages 13 and up. Let $a$ be a person's age. Which statement best describes the solutions to the inequality for this rule?
Answer: The inequality is $a\ge 13$ , and it has infinitely many solutions: any age 13 or older. This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct statement is B: a ≥ 13 with infinitely many solutions (any age 13 or older). Errors claim only one solution (choice A), reverse to ≤ (choice C), or say finite solutions (choice D). Context: 'at least 13' means 13, 14, 15, ... all valid (infinite possible ages). Mistakes include treating it like an equation or not recognizing infinity of solutions.
To join an online class, your score must be more than 75 on a placement quiz. Let $x$ be your score. Which inequality and number line graph are correct?
Answer: $x>75$ ; open circle at 75, shade right This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct choice is B: x > 75 with open circle at 75 and shading right, excluding 75 but including all greater scores. Errors include using ≥ with closed circle (choice A, including 75), reversing to < or ≤ (choices C and D), or wrong shading direction. To write and graph: (1) identify 'more than' as excluding boundary, (2) translate to >, (3) write x > 75; for graph, use open ○ at 75 and shade right. Infinite solutions exist, like all reals > 75; common mistakes are confusing > with ≥ or shading left.
A school club requires members to be at least 13 years old. Let $a$ be a student's age (in years). Which inequality correctly represents this rule?
Answer: $a\ge 13$ This question tests writing inequalities like a > c or a ≥ c from real-world constraints, understanding that inequalities have infinitely many solutions, and representing them on number lines with proper boundary markers such as closed circles (●) for inclusive or open circles (○) for exclusive, along with shading in the correct direction. Constraint words translate to symbols: 'at least' means ≥ which includes the boundary (a ≥ 13 means 13 or more), 'at most' means ≤ which includes it, 'more than' means > which excludes the boundary, and 'less than' means < which excludes it; inequalities have infinite solutions since they are satisfied by infinitely many numbers, unlike equations with finite solutions. For example, 'must be at least 13 years old' translates to a ≥ 13 where a is age, with solutions like 13, 14, 15, and infinitely many more, graphed as a number line with a closed circle at 13 and shading to the right (●======>). The correct inequality here is a ≥ 13, which includes 13 and all greater ages, matching choice B. A common error is using > instead of ≥ for 'at least,' excluding 13 incorrectly, or reversing to < or ≤, or misunderstanding infinite solutions by thinking only whole numbers count when actually all real numbers ≥13 satisfy it. To write this: (1) identify 'at least' as inclusive greater, (2) translate to ≥, (3) write a ≥ 13. For graphing: (1) draw number line, (2) mark 13, (3) use closed ● for ≥, (4) shade right for greater values; remember infinite solutions include all reals ≥13, and in context, ages like 13.5 would theoretically satisfy though practically ages are whole; mistakes include wrong symbol or circle type.
A theme park ride requires riders to be at least 48 inches tall. Let $h$ be a rider's height in inches. Which value is a solution to the inequality?
Answer: $h=48$ This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct solution is h = 48 (choice B), as h ≥ 48 includes 48 and taller. Errors are values below 48 like 47, 30, or 0 (choices A, C, D), not satisfying ≥ 48. To check: (1) identify 'at least' as ≥, (2) write h ≥ 48, (3) test values (48 works, 47 does not). Infinite solutions: all reals ≥ 48; mistakes treat it as > or pick non-solutions.
The graph shows the solution to an inequality on a number line. Which inequality matches this representation?
Answer: $x \geq 4$ because the circle is closed and shading goes right The number line shows a closed circle at 4 with shading extending to the right, indicating that 4 is included and all numbers greater than 4 are solutions. This represents $x ≥ 4$ . Choice A has the wrong direction. Choice C uses an open circle (strict inequality) but the diagram shows closed. Choice D contradicts itself about whether 4 is included.
A school club says you must sell at least 10 raffle tickets to earn a prize. Let $x$ be the number of tickets you sell. Which inequality correctly represents this rule?
Answer: $x\ge 10$ This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct inequality is x ≥ 10, which matches choice C, representing selling at least 10 tickets, including exactly 10. A common error is using > instead of ≥ for 'at least,' excluding the boundary (like choice A), or reversing the inequality to ≤ or < (choices B and D), which would mean selling 10 or fewer. To write this: (1) identify 'at least' as including the boundary, (2) translate to ≥, (3) write x ≥ 10. Remember, this has infinitely many solutions, like all integers ≥ 10, and in real numbers, uncountably many, such as 10.5, 11.2, etc.
A teacher says, "Your missing assignments must be less than 3 to go on the field trip." Let $x$ be the number of missing assignments. Which set lists only solutions to the inequality?
Answer: $\{0,1,2\}$ This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct set is A: {0,1,2}, all solutions to x < 3 (less than 3, excluding 3). Errors include sets with 3 or higher (choices B, C, D), which are not solutions. To identify: (1) translate 'less than 3' to x < 3, (2) check sets (only A has all < 3). Infinite solutions overall, but question asks for set listing only solutions; mistakes pick sets including non-solutions.
A movie theater has a rule: "Children's tickets cost $8, and you can buy at most $40 worth of children's tickets." Let $x$ be the total amount spent (in dollars) on children's tickets. Which option correctly gives the inequality and a correct interpretation of its solutions?
Answer: $x\le 40$ ; any amount up to and including $40 works (infinitely many possible dollar amounts). This question tests writing inequalities like x > c or x < c from real-world constraints, understanding that they have infinitely many solutions, and representing them on number lines with proper boundary markers (● for included, ○ for excluded) and shading. Constraint words translate to symbols: 'at least' means ≥ (includes), 'at most' means ≤ (includes boundary, like x ≤ 40 allows 40 or less), 'more than' means > (excludes), 'less than' means < (excludes); inequalities have infinite solutions unlike equations, and on a number line, use closed ● for ≥/≤, open ○ for >/<, shading right for greater and left for lesser. For example, 'at most 40' gives x ≤ 40 with infinite solutions up to 40. The correct is x ≤ 40 with interpretation of any amount up to including 40, infinitely many. A common error is using <, excluding 40. Infinite solutions: uncountably many reals ≤40. Mistakes: wrong symbol, not recognizing infinity.
A teacher says, "To be on the honor roll, your average must be more than 90." Let $m$ be the average. Which is the correct interpretation of $m>90$ in this context?
Answer: Any average greater than 90 counts, but 90 does not. This question tests interpreting inequalities like m > 90 from 'more than,' understanding it excludes boundary and has infinite solutions. 'More than' → > excludes (m > 90: above 90, not 90); infinite solutions: 90.1,91,... . Example: 'more than 90' → m > 90, interpretation: greater than 90 counts, 90 does not. Correct interpretation is any average greater than 90 counts but 90 does not, matching choice C. Error: thinking includes 90 like ≥, or reversing to less. For interpretation: recognize exclusive greater; infinite qualifying averages. Context: 90 doesn't qualify; mistakes: including boundary, misreading symbol.
A temperature sensor triggers an alarm when the temperature exceeds 75°F. The sensor recorded temperatures of 73°F, 76°F, 75°F, and 78°F. Based on the number line diagram shown, which inequality correctly represents when the alarm will sound?
Answer: $t > 75$ where $t$ is temperature in degrees Fahrenheit The alarm triggers when temperature 'exceeds' 75°F, meaning it goes above 75°F but not including 75°F itself. This is represented by $t > 75$ . The number line shows an open circle at 75 and shading to the right, confirming values greater than 75. Choice B uses ≥, which would include 75°F. Choices C and D represent temperatures below or equal to 75°F, which is when the alarm would NOT sound.
A snack bar has a sign: "Buy more than 6 bottles of water to get a discount." Let $x$ be the number of bottles you buy. Which description correctly explains why the inequality has infinitely many solutions?
Answer: Because $x>6$ is true for $x=7,8,9,10,\dots$ , and there is no largest number of bottles you could buy. This question tests writing inequalities like x > c or x < c from real-world constraints, understanding infinitely many solutions, and representing on number lines with proper boundary markers (●/○) and shading. Constraint words translate to symbols: 'at least' → ≥ (includes boundary: x ≥ 10 means 10 or more), 'at most' → ≤ (includes: x ≤ 50 means 50 or less), 'more than' → > (excludes boundary: x > 75 means above 75, not including 75), 'less than' → < (excludes: x < 60 below 60); inequalities have infinitely many solutions (x > 10 includes 11, 12, 13, ..., 1000, ...—continues forever, unlike equations with one solution); on a number line, mark boundary at c (for x > 10, mark 10), use closed ● if ≥ or ≤ (includes), open ○ if > or < (excludes), shade right for > / ≥ (greater values), left for < / ≤ (lesser values). For example, 'must have at least 10 tickets' → x ≥ 10 (x = tickets, 10 or more), solutions: 10, 11, 12, ... infinitely many (all non-negative integers ≥ 10), graph: ● at 10 shaded right (●======> ), includes 10 and all greater; or 'score more than 75' → x > 75 (above 75, excludes 75), graph: ○======> at 75 (open circle, shade right), solutions: 76, 77, 78, ... infinite. The correct description is B, explaining x > 6 has infinitely many solutions like 7,8,9,... with no upper limit. Errors claim one solution (choice A), finite solutions (choice C), or no solutions (choice D). To understand: (1) identify 'more than' as >, (2) write x > 6, (3) recognize infinite integers > 6. Infinite solutions are uncountable in reals > 6; mistakes treat as finite or like equations.
A parking meter accepts coins only if the total value is greater than $0.75. Sarah has quarters ($0.25 each) and wants to determine how many quarters $q$ she needs. Which inequality and solution representation is correct?
Answer: $0.25q > 0.75$ ; Sarah needs more than 3 quarters, so at least 4 quarters The meter requires a value greater than $0.75, so $0.25q > 0.75$ . Solving: $q > 3$ . Since Sarah can't use a fraction of a quarter, she needs at least 4 quarters. Choice B uses ≥ and incorrectly concludes 3 quarters is enough. Choice C uses the wrong inequality direction. Choice D omits the 0.25 coefficient and gives a nonsensical interpretation.

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