Solve and Graph Linear Inequalities - 7th Grade Math

1. Solve the inequality $-2x + 1 > 9$.

   $x < -4$. Subtract 1, divide by -2 (flip sign): $-2x > 8$, so $x < -4$.

2. Solve the inequality $3x + 5 > 20$.

   $x > 5$. Subtract 5, then divide by 3: $3x > 15$, so $x > 5$.

3. What inequality represents: "You have $20$ dollars and spend $3x$; you want more than $5$ left"?

   $20-3x>5$. Money left is $20-3x$; "more than $5$" means $>5$.

4. What inequality symbol means "greater than" when translating a word problem?

   $>$. The symbol $>$ indicates one value exceeds another.

5. Identify the solution: Solve $\frac{3}{4}x-2\ge 1$.

   $x\ge 4$. Add $2$, then multiply by $\frac{4}{3}$: $\frac{3}{4}x\ge 3$.

6. What does the inequality symbol $>$ mean in a word problem?

   $>$ means "greater than.". This symbol indicates one value is larger than another.

7. What must you do to an inequality sign when you multiply or divide by a negative number?

   Reverse the sign: $>$ becomes $<$ and $<$ becomes $>$. Multiplying/dividing by negatives flips the inequality direction.

8. Solve the inequality $\frac{1}{2}x + 3 < 7$.

   $x < 8$. Subtract 3, multiply by 2: $\frac{1}{2}x < 4$, so $x < 8$.

9. Which graph uses an open circle: $x>3$ or $x\ge 3$?

   $x>3$ uses an open circle at $3$. Strict inequalities use open circles; $\ge$ would use closed.

10. Which type of point is used on a number line for $x < -1$: open or closed?

    Open circle at $-1$. Open circle shows -1 is not included in the solution set.

11. Identify the solution: Solve $5x-7<8$.

    $x<3$. Add $7$, then divide by $5$: $5x<15$.

12. Identify the solution: "A gym charges $10$ plus $4$ dollars per visit; total cost is less than $30$."

    $x<5$. Set up $10+4x<30$, then solve: $4x<20$.

13. Which graph shows a strict inequality, $x>3$ or $x<3$: open circle or closed circle?

    Open circle. Open circles exclude the boundary value in strict inequalities.

14. Which value satisfies $4x+1<9$: $x=1$ or $x=3$?

    $x=1$. Test: $4(1)+1=5<9$ ✓; $4(3)+1=13>9$ ✗.

15. What is the solution for hours $h$ in $10+2.5h<40$?

    $h<12$. Subtract 10, divide by 2.5: $2.5h<30$, so $h<12$.

16. Solve the inequality $-\frac{3}{4}x + 2 \le 5$.

    $x \ge -4$. Subtract 2, divide by $-\frac{3}{4}$ (flip sign): $x \ge -4$.

17. What inequality represents: "After spending $7.50$, you have less than $12$ left from $m$ dollars"?

    $m-7.5<12$. Starting amount minus spending must be less than 12.

18. What is the graph description for $x<-2$: circle type and shading direction?

    Open circle at $-2$, shade left. For $x<-2$, exclude -2 (open circle) and shade values less than -2.

19. Which value is a solution to $2x + 1 > 9$: $x = 3$ or $x = 5$?

    $x = 5$. Test: $2(5) + 1 = 11 > 9$ ✓; $2(3) + 1 = 7 \not> 9$ ✗.

20. What inequality represents: "You need more than $60$ points; you already have $18$ and earn $6$ per game $g$"?

    $18+6g>60$. Current points plus points per game must exceed 60.

21. What is the solution to the inequality $-\frac{3}{4}x-2<1$?

    $x>-4$. Add 2, divide by $-\frac{3}{4}$ (flip sign): $-\frac{3}{4}x<3$, so $x>-4$.

22. Identify the inequality phrase: Which symbol matches "no more than"?

    $\le$. "No more than" means less than or equal to.

23. Which type of point is used on a number line for $x > 5$: open or closed?

    Open circle at $5$. Open circle shows 5 is not included in the solution set.

24. Which graph uses a closed circle: $x< -2$ or $x\le -2$?

    $x\le -2$ uses a closed circle at $-2$. "Less than or equal to" includes the endpoint.

25. What is the solution for games $g$ in $18+6g>60$?

    $g>7$. Subtract 18, divide by 6: $6g>42$, so $g>7$.

26. Solve the inequality $4x - 7 < 9$.

    $x < 4$. Add 7, then divide by 4: $4x < 16$, so $x < 4$.

27. Write an inequality for: "Twice a number $x$ plus $3$ is greater than $11$."

    $2x + 3 > 11$. "Twice" means $2x$; "plus" means add; "greater than" is $>$.

28. What inequality represents: "$7$ less than twice a number $x$ is at most $9$"?

    $2x-7\le 9$. "Less than twice" is $2x-7$; "at most" means $\le$.

29. What inequality symbol matches the phrase "at least $r$"?

    Use $\ge r$. "At least" means the value can equal or exceed $r$.

30. What inequality symbol matches the phrase "no more than $r$"?

    Use $\le r$. "No more than" means the value cannot exceed $r$.

31. Identify the inequality phrase: Which symbol matches "at least"?

    $\ge$. "At least" means greater than or equal to.

32. What inequality symbol means "less than" when translating a word problem?

    $<$. The symbol $<$ indicates one value is smaller than another.

33. What is the solution to $m-7.5<12$?

    $m<19.5$. Add 7.5 to both sides: $m<12+7.5=19.5$.

34. What does the inequality symbol $<$ mean in a word problem?

    $<$ means "less than.". This symbol indicates one value is smaller than another.

35. What inequality represents: "A number $x$ increased by $5$ is greater than $12$"?

    $x+5>12$. "Increased by" means addition; "greater than" gives $>$.

36. What does the solution set of an inequality represent in a word problem context?

    All values of $x$ that make the situation true. The solution set includes all possible values that satisfy the inequality.

37. What is the solution to the inequality $-2x+1>9$?

    $x<-4$. Subtract 1, divide by -2 (flip sign): $-2x>8$, so $x<-4$.

38. Graph the solution on a number line: $x\ge 2$ (open or closed circle, and shading direction)?

    Closed circle at $2$; shade to the right. "Greater than or equal" includes $2$ and all values to the right.

39. What is the solution to the inequality $5x-4<16$?

    $x<4$. Add 4, divide by 5: $5x<20$, so $x<4$.

40. What inequality represents: "Three times a number $x$ decreased by $2$ is less than $10$"?

    $3x-2<10$. "Times" means multiply; "decreased by" means subtract; "less than" gives $<$.

41. Write an inequality for: "A number $x$ decreased by $6$ is less than $10$."

    $x - 6 < 10$. "Decreased by" means subtraction; "less than" is $<$.

42. Solve and interpret: "A gym costs $10 plus $3 per visit. You can spend less than $25. Find $x$."

    $10 + 3x < 25$ so $x < 5$. Total cost must be under $25: $10 + 3x < 25$, so $x < 5$ visits.

43. Identify the inequality phrase: Which symbol matches "more than"?

    $>$. "More than" indicates strict inequality (not equal).

44. Identify the solution: Solve $3x+2>11$.

    $x>3$. Subtract $2$, then divide by $3$: $3x>9$.

45. Solve and interpret: "You have $15 and spend $2 per snack. You want more than $5 left. Find $x$."

    $15 - 2x > 5$ so $x < 5$. Money left after spending: $15 - 2x > 5$, so $x < 5$ snacks.

46. Identify the solution: Solve $\frac{1}{2}x+3<7$.

    $x<8$. Subtract $3$, then multiply by $2$: $\frac{1}{2}x<4$.

47. Which value is a solution to $2x+1<9$: $x=3$ or $x=5$?

    $x=3$. Test: $2(3)+1=7<9$ is true; $2(5)+1=11<9$ is false.

48. What is the solution to the inequality $\frac{1}{2}x+3<7$?

    $x<8$. Subtract 3, multiply by 2: $\frac{1}{2}x<4$, so $x<8$.

49. What happens to an inequality when you multiply or divide both sides by a negative number?

    The inequality sign reverses direction. Multiplying/dividing by negatives flips the inequality.

50. What inequality represents: "A number $x$ increased by $5$ is more than $12$"?

    $x+5>12$. "Increased by" means addition; "more than" means $>$.

51. What is the solution to the inequality $3x+2>11$?

    $x>3$. Subtract 2, divide by 3: $3x>9$, so $x>3$.

52. What inequality represents: "The cost is a $10$ dollar fee plus $2.5$ dollars per hour $h$, under $40$"?

    $10+2.5h<40$. Fixed fee plus hourly rate must be under the limit.

53. Identify the solution: Solve $-2x+1>9$.

    $x<-4$. Subtract $1$, divide by $-2$, and flip the sign: $-2x>8$.

54. Graph the solution set of $x < 2$ on a number line: which direction is shaded?

    Shade left from an open circle at $2$. For $x < 2$, shade all values less than 2 (to the left).

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