Inequalities & Absolute Value - ACT Math

1. What is the solution to $5-2x<1$?

   $x>2$. Subtract $5$, then divide by $-2$ (reverse inequality).

2. What is the definition of absolute value $|a|$ for real $a$?

   $|a|=a\text{ if }a\ge 0;\ |a|=-a\text{ if }a<0$. Distance from zero; always non-negative.

3. What is the distance between $-2$ and $5$ on the number line written with absolute value?

   $|-2-5|$. Distance formula uses absolute value of difference.

4. What happens to an inequality when you add the same number to both sides?

   The inequality direction stays the same. Addition preserves the inequality direction.

5. What is $|5|$?

   $5$. Absolute value of positive number equals itself.

6. What is the solution to $x^2 \leq 16$?

   $-4 \leq x \leq 4$. Take square root: $-4 \leq x \leq 4$.

7. Solve the inequality: $2x + 3 > 7$.

   $x > 2$. Subtract 3 from both sides, then divide by 2.

8. What is the solution set for $|x| < 4$?

   $-4 < x < 4$. Distance from 0 is less than 4.

9. What does the inequality symbol $\leq$ mean?

   Less than or equal to. Symbol combines less than with equality.

10. Solve for $x$: $3x - 4 \neq 5$.

    $x \neq 3$. Solve $3x - 4 = 5$ gives $x = 3$, so $x \neq 3$.

11. What is the numerical distance between $-2$ and $5$ on the number line?

    $7$. Calculate $|-2-5| = |-7| = 7$.

12. What is the property of $|a + b|$?

    $|a + b| \leq |a| + |b|$. Triangle inequality property for absolute values.

13. Find $x$ if $|3x + 4| = 7$.

    $x = 1$ or $x = -\frac{11}{3}$. Set $3x + 4 = \pm 7$ and solve each equation.

14. What is the solution to $|x| \geq 6$?

    $x \leq -6$ or $x \geq 6$. Distance from 0 is at least 6 units.

15. Solve the inequality: $4x - 5 \geq 3$.

    $x \geq 2$. Add 5 to both sides, then divide by 4.

16. What is the solution to $|x|\ge 6$?

    $x\le -6\text{ or }x\ge 6$. Distance from zero is at least $6$.

17. What is the solution to $|x|>2$?

    $x<-2\text{ or }x>2$. Distance from zero is greater than $2$.

18. Find $x$ if $|2x - 1| = 3$.

    $x = 2$ or $x = -1$. Set $2x - 1 = \pm 3$ and solve each case.

19. What is $|0|$?

    0. Absolute value of zero is always zero.

20. What is the solution set of $x<5$ written in interval notation?

    $(-\infty,5)$. Use parenthesis for strict inequality (less than, not equal to).

21. What is the solution to $|x-4|<0$?

    No solution. Absolute value cannot be negative.

22. What inequality corresponds to the interval $[-3,7)$?

    $-3\le x<7$. Bracket includes $-3$, parenthesis excludes $7$.

23. What is the solution to $|x|<4$?

    $-4<x<4$. Distance from zero is less than $4$.

24. What is the solution to $2x+1\le 9$?

    $x\le 4$. Subtract $1$ from both sides, then divide by $2$.

25. Solve for $x$: $|x + 2| \geq 3$.

    $x \leq -5$ or $x \geq 1$. Distance from -2 is at least 3 units.

26. What is $|-7|$?

    $7$. Absolute value of negative number equals its positive value.

27. What is the solution set of $x\le 4$ written in interval notation?

    $(-\infty,4]$. Use bracket for inclusive inequality (less than or equal to).

28. What is the solution to $3x-5>7$?

    $x>4$. Add $5$ to both sides, then divide by $3$.

29. Solve for $x$: $2 < 3x + 1 \leq 8$.

    $\frac{1}{3} < x \leq \frac{7}{3}$. Subtract 1, then divide by 3 for each part.

30. What inequality corresponds to the interval $(-1,2]$?

    $-1<x\le 2$. Parenthesis excludes $-1$, bracket includes $2$.

31. What is the solution to $x^2 > 9$?

    $x > 3$ or $x < -3$. Take square root of both sides, considering both signs.

32. What is the solution set of $x\ge -2$ written in interval notation?

    $[-2,\infty)$. Use bracket for inclusive inequality (greater than or equal to).

33. What is the solution to $x\ne 4$ in interval notation?

    $(-\infty,4)\cup(4,\infty)$. Not equal excludes the single value from all real numbers.

34. What is the solution to $\frac{x-3}{2}<5$?

    $x<13$. Multiply by $2$, then add $3$ to both sides.

35. What is the solution to $|x-3|=0$?

    $x=3$. Absolute value equals zero only when expression equals zero.

36. Solve the inequality: $x + 5 \leq 9$.

    $x \leq 4$. Subtract 5 from both sides to isolate $x$.

37. What is the inequality for 'at most 10'?

    $x \leq 10$. 'At most' means less than or equal to.

38. What is the solution to $|3x+6|<9$?

    $-5<x<1$. Solve $-9 < 3x+6 < 9$, then divide by $3$.

39. Solve for $x$: $|x - 3| = 5$.

    $x = 8$ or $x = -2$. Distance from 3 equals 5, so $x-3 = \pm 5$.

40. What is the solution to $|x-2|=7$?

    $x=-5\text{ or }x=9$. Distance from $2$ equals $7$; solve $x-2 = \pm 7$.

41. Solve for $x$: $|x - 4| < 2$.

    $2 < x < 6$. Distance from 4 is less than 2 units.

42. What is the solution set for $|x| = 0$?

    $x = 0$. Only zero has absolute value of zero.

43. Solve for $x$: $|x + 7| > 4$.

    $x > -3$ or $x < -11$. Distance from -7 exceeds 4 units.

44. What is $|0|$?

    $0$. Absolute value of zero equals zero.

45. What is the solution to $|x-4|\ge 0$?

    All real numbers. Absolute value is always non-negative.

46. What is the solution to $|x-5|<2$?

    $3<x<7$. Distance from $5$ is less than $2$.

47. Solve the inequality: $5 - 2x < 1$.

    $x > 2$. Rearrange to $-2x < -4$, then divide by -2.

48. What is the solution to $x^2 \leq 16$?

    $-4 \leq x \leq 4$. Take square root: $-4 \leq x \leq 4$.

49. What is the solution to $|x-1|\le 0$?

    $x=1$. Only when absolute value equals zero.

50. What is the solution to $-4 < x < 2$?

    All $x$ between -4 and 2. Compound inequality defines interval notation.

51. What is the solution to $|2x+3|=1$?

    $x=-2\text{ or }x=-1$. Solve $2x+3 = 1$ and $2x+3 = -1$.

52. Solve the inequality: $-3x + 4 > 1$.

    $x < 1$. Rearrange to $-3x > -3$, then divide by -3.

53. What is the graphical representation of $|x|$ on a number line?

    Distance from 0. Absolute value measures distance from origin.

54. What does the inequality symbol $>$ mean?

    Greater than. Symbol for strict greater than comparison.

55. Solve the inequality: $x + 5 \leq 9$.

    $x \leq 4$. Subtract 5 from both sides to isolate $x$.

56. What compound inequality corresponds to the interval $[2,5]$?

    $2\le x\le 5$. Closed interval creates an 'and' compound inequality.

57. What is the result of $|-9|$?

    9. Absolute value converts negative to positive.

58. State the property: $|a \times b|$.

    $|a| \times |b|$. Absolute value of product equals product of absolute values.

59. What is the absolute value of -12?

    12. Absolute value makes negative numbers positive.

60. What is the solution to the compound inequality $-1\le 2x+3<7$?

    $-2\le x<2$. Subtract $3$, divide by $2$, then solve each part.

61. What compound inequality corresponds to $(-\infty,1)\cup(4,\infty)$?

    $x<1\text{ or }x>4$. Union of two separate intervals creates an 'or' compound inequality.

62. What is the absolute value of 0?

    0. Zero has no sign, so absolute value is zero.

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

    The inequality sign reverses direction. Multiplying by negative values reverses the inequality direction.

64. Describe the inequality $x > 5$.

    All $x$ greater than 5. Open interval extending infinitely rightward.

65. What does the inequality symbol $< $ mean?

    Less than. Symbol for strict less than comparison.

66. What is the solution to $-4x\ge 12$?

    $x\le -3$. Divide by $-4$ and reverse the inequality sign.

67. What is the inequality for 'at least 15'?

    $x \geq 15$. 'At least' means greater than or equal to.

68. What is the solution to $|x+1|\le 3$?

    $-4\le x\le 2$. Distance from $-1$ is at most $3$.

69. Solve for $x$: $x^2 - 4x < 0$.

    $0 < x < 4$. Factor as $x(x-4) < 0$, find sign intervals.

70. What is the solution to $|2x-1|\ge 5$?

    $x\le -2\text{ or }x\ge 3$. Solve $2x-1 \le -5$ or $2x-1 \ge 5$.

71. What is the solution to $|x|\le 3$?

    $-3\le x\le 3$. Distance from zero is at most $3$.

72. What is the solution to the compound inequality $3<\frac{x+1}{2}\le 5$?

    $5<x\le 9$. Subtract $1$, multiply by $2$, then solve each part.

73. What does the inequality symbol $\geq$ mean?

    Greater than or equal to. Symbol combines greater than with equality.

74. What is the solution set of $x>3$ written in interval notation?

    $(3,\infty)$. Use parenthesis for strict inequality (greater than, not equal to).

75. What is the absolute value of -7?

    7. Absolute value converts negative numbers to positive.

76. What inequality symbol represents 'greater than'?

    $>$. Standard symbol for strict inequality.

77. What is the solution to $|x| > 3$?

    $x > 3$ or $x < -3$. Distance from 0 exceeds 3 units.

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