Linear Inequalities - PSAT Math

1. What is the interval notation for the solution set of $x > 3$?

   $(3, \infty)$. Open parenthesis excludes 3; infinity always uses parenthesis.

2. What is the compound inequality for the interval $[1,4)$?

   $1\le x<4$. Square bracket includes 1, parenthesis excludes 4.

3. Which inequality matches the statement "$x$ is greater than or equal to $-3$"?

   $x\ge -3$. This directly translates the given statement.

4. What is the solution set in interval notation for $x\le -1$ or $x>3$?

   $(-\infty,-1]\cup(3,\infty)$. Union symbol combines two disjoint intervals.

5. Which values satisfy both inequalities $x > -1$ and $x \le 3$?

   $(-1,3]$. Find the intersection where both conditions are true.

6. Solve the compound inequality $-2\le 3x+1<7$.

   $-1\le x<2$. Subtract 1, divide by 3 throughout the compound inequality.

7. What is the solution to the compound inequality $-2 \le x + 1 < 5$?

   $-3 \le x < 4$. Subtract 1 from all parts: $-3 \le x < 4$.

8. What is the solution to the inequality $3x-7\le 5$?

   $x\le 4$. Add 7 to both sides: $3x \le 12$, then divide by 3.

9. What is the solution to the inequality $x+5<12$?

   $x<7$. Subtract 5 from both sides: $12-5=7$.

10. Which value is a solution to $3x-5\le 7$: $x=3$ or $x=5$?

    $x=3$. Check: $3(3)-5=4\le 7$ is true; $3(5)-5=10\le 7$ is false.

11. Identify the correct graph endpoint type for $x\le 5$: open circle or closed circle at $5$?

    Closed circle at $5$. Closed circle includes the endpoint value in the solution.

12. Identify the correct solution set for $|x|<4$ written as a compound inequality.

    $-4<x<4$. Absolute value less than splits into two inequalities.

13. Which values satisfy $x < -2$ or $x \ge 3$ in interval notation?

    $(-\infty, -2) \cup [3, \infty)$. Union symbol $\cup$ combines two separate intervals.

14. What is the graph symbol for $x \ge a$ on a number line?

    Closed circle at $a$; shade to the right. Closed circle includes the endpoint; shade shows valid values.

15. What is the solution to the compound inequality $-1 < x + 2 \le 5$?

    $-3 < x \le 3$. Subtract 2 from all parts of the compound inequality.

16. What is the solution to the inequality $5 - x \ge 1$?

    $x \le 4$. Rearrange: $-x \ge 1 - 5$, then multiply by -1 and flip.

17. Identify whether the boundary point is included for $x\ge 4$: open circle or closed circle?

    Closed circle at $4$. Closed circle includes the boundary point in the solution set.

18. What is the solution to the inequality $4(2-x)<8$?

    $x>0$. Distribute, rearrange, then divide by $-4$ (flip sign).

19. What is the solution to the inequality $2x-7>5$?

    $x>6$. Add 7, then divide by 2: $2x > 12$, so $x > 6$.

20. Which graph endpoint is used for $x\le a$: open circle or closed circle at $a$?

    Closed circle at $a$. Closed circle includes the endpoint value.

21. What interval notation corresponds to $x \ge 3$?

    $[3,\infty)$. Square bracket includes 3; infinity is always with parenthesis.

22. What inequality symbol results after multiplying both sides by a negative number?

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

23. What does the phrase "no more than" translate to in an inequality?

    $\le$. "No more than" sets a maximum, so less than or equal to.

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

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

25. Identify the solution: Solve $3x-5 \le 10$.

    $x \le 5$. Add 5: $3x \le 15$, then divide by 3.

26. What is the solution to the inequality $\frac{x}{4} - 2 \ge 3$?

    $x \ge 20$. Add 2, multiply by 4: $x \ge 5 \times 4$.

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

    The inequality sign reverses direction. This preserves the inequality relationship when changing sign.

28. What is the solution to the inequality $2x - 3 \ge 9$?

    $x \ge 6$. Add 3, then divide by 2: $2x \ge 12$, so $x \ge 6$.

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

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

30. What happens to an inequality when you multiply both sides by a negative number $k<0$?

    Reverse the inequality direction. Multiplying by a negative number flips the inequality sign.

31. What interval notation represents the solution set of $x < -1$?

    $(-\infty,-1)$. Parenthesis excludes $-1$, extends to negative infinity.

32. What is the boundary line equation when graphing $y>2x-3$ in the coordinate plane?

    $y=2x-3$. The boundary line is where the inequality becomes an equation.

33. What inequality matches the interval notation $(-\infty,5)$?

    $x < 5$. Parenthesis at $5$ means $5$ is not included in the solution.

34. What interval notation represents the inequality $-3<x\le 5$?

    $(-3,5]$. Parenthesis excludes $-3$; bracket includes $5$.

35. What is the solution to the absolute value inequality $|x|<5$?

    $-5<x<5$. Distance from 0 is less than 5.

36. What is the interval notation for $x\ge a$?

    $[a,\infty)$. Bracket includes $a$; extends to positive infinity.

37. What is the solution to the inequality $-3x+4\ge 10$?

    $x\le -2$. Subtract $4$, divide by $-3$, and flip the inequality.

38. What interval notation matches the inequality $x>-2$?

    $(-2,\infty)$. Parenthesis excludes -2, extends to infinity.

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

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

40. What is the solution set meaning of $x>3$ on a number line?

    All real numbers greater than $3$. Open circle at 3, shading extends rightward infinitely.

41. What is the solution to the compound inequality $3 < 2x + 1 \le 9$?

    $1 < x \le 4$. Subtract 1, then divide by 2: $2 < 2x \le 8$.

42. Solve the inequality $3x-5\le 7$.

    $x\le 4$. Add 5 to get $3x \le 12$, then divide by 3.

43. Identify whether the boundary point is included for $x<4$: open circle or closed circle?

    Open circle at $4$. Open circle excludes the boundary point from the solution set.

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

    $(3,\infty)$. Open parenthesis at 3 since x is strictly greater than 3.

45. What is the solution to the inequality $-\frac{3}{2}x\ge 6$?

    $x\le -4$. Divide by $-\frac{3}{2}$ (multiply by $-\frac{2}{3}$), flip sign.

46. What is the solution to the compound inequality $x<-1$ or $x\ge 3$?

    All $x$ with $x<-1$ or $x\ge 3$. Values less than -1 OR greater than/equal to 3.

47. What is the solution to the inequality $4(x-2)<12$?

    $x<5$. Distribute 4, then solve: $4x-8<12$, so $4x<20$.

48. Find and correct the error: From $-4x \le 8$ a student wrote $x \le -2$.

    Correct: $x \ge -2$. Student forgot to flip the sign when dividing by -4.

49. What does a closed circle at $x=a$ mean on a number line graph of an inequality?

    $a$ is included (inequality is $\le$ or $\ge$). Closed circles represent inequalities that include equality.

50. Identify the solution: Solve $2 \le x+1 < 7$.

    $1 \le x < 6$. Subtract 1 from all parts: $1 \le x < 6$.

51. What is the solution set in interval notation for $-2\le x<5$?

    $[-2,5)$. Includes $-2$, excludes $5$.

52. What is the solution to the inequality $4(x-2) > 8$?

    $x > 4$. Distribute: $4x - 8 > 8$, then $4x > 16$, so $x > 4$.

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

    $x \ge -12$. Multiply both sides by 4.

54. What is the graphing rule for $y>mx+b$ versus $y\ge mx+b$ on a coordinate plane?

    $>$ dashed boundary; $\ge$ solid boundary (shade above). Strict inequalities use dashed lines; non-strict use solid lines.

55. What interval notation represents the solution set of $x \ge 2$?

    $[2,\infty)$. Square bracket includes 2, extends to positive infinity.

56. Identify the graph boundary type for $y>2x-1$: solid line or dashed line?

    Dashed line. Strict inequalities ($>$ or $<$) use dashed boundaries.

57. Which endpoint symbol is used on a number line for $x<a$ or $x>a$: open or closed?

    Open circle. Open circle indicates the endpoint is not included.

58. Solve the inequality $2(3x-1)\le 4x+7$.

    $x\le \frac{9}{2}$. Expand, combine like terms, then isolate $x$.

59. Identify the solution set in interval notation for $5-2x\ge 1$.

    $(-\infty,2]$. Rearrange: $-2x \ge -4$, so $x \le 2$.

60. What is the solution set of $\frac{x}{3}-2>1$?

    $x>9$. Add 2 to get $\frac{x}{3}>3$, then multiply by 3.

61. Find the slope-intercept form of $2x+3y\le 6$.

    $y\le -\frac{2}{3}x+2$. Isolate $y$: subtract $2x$, divide by $3$.

62. What is the solution to the inequality $\frac{x}{4}\le -3$?

    $x\le -12$. Multiply both sides by 4: $4×(-3)=-12$.

63. What is the solution to the absolute value inequality $|x-2| \le 5$?

    $-3 \le x \le 7$. $|x-2| \le 5$ means $-5 \le x-2 \le 5$, so add 2.

64. Identify the correct result of multiplying both sides of $x<-3$ by $-2$.

    $-2x>6$. Multiply by -2 and flip: $(-2)(x) > (-2)(-3)$.

65. Identify the solution: Solve $-2x+7 > 1$.

    $x < 3$. Subtract 7: $-2x > -6$, divide by -2 and flip sign.

66. What inequality corresponds to the interval $(-3,4]$?

    $-3 < x \le 4$. Parenthesis excludes $-3$, bracket includes 4.

67. What is the solution to the inequality $\frac{x-1}{2}>3$?

    $x>7$. Multiply by $2$, then add $1$ to isolate $x$.

68. What is the interval notation for $-2<x\le 4$?

    $(-2,4]$. Parenthesis excludes -2, bracket includes 4.

69. Identify the correct graph direction for $x< -1$: shade left or shade right of $-1$?

    Shade left of $-1$. Values less than -1 are to the left on a number line.

70. Which value satisfies $2x+1\le 7$: $x=2$ or $x=4$?

    $x=2$. Test: $2(2)+1=5 \le 7$ ✓; $2(4)+1=9 > 7$ ✗.

71. What happens to an inequality when you subtract the same number $c$ from both sides?

    The solution set is unchanged. Subtraction, like addition, preserves the inequality relationship.

72. Identify the solution set for $x < 1$ and $x \ge -2$ in interval notation.

    $[-2, 1)$. Intersection of $x < 1$ and $x \ge -2$ gives the overlap.

73. What is the solution to the inequality $-2x>8$?

    $x<-4$. Divide by -2 and flip the sign: $x < 8/(-2)$.

74. What is the compound inequality equivalent to the interval $[0,5)$?

    $0\le x<5$. Bracket includes 0, parenthesis excludes 5.

75. What is the interval notation for $x\ge 5$?

    $[5,\infty)$. Bracket includes 5; infinity always uses parenthesis.

76. What happens to an inequality sign when you add or subtract the same number on both sides?

    The inequality sign stays the same. Addition and subtraction preserve inequality direction.

77. Find and correct the error: From $-2x>6$ a student wrote $x> -3$.

    Correct: $x<-3$. Student forgot to flip the sign when dividing by -2.

78. What is the solution to $7x+3<2x+18$?

    $x<3$. Subtract $2x$ and 3 from both sides: $5x < 15$, divide by 5.

79. Identify the solution set of $x<1$ or $x\ge 4$ in interval notation.

    $(-\infty,1)\cup[4,\infty)$. Union symbol combines two separate intervals.

80. What is the solution to the inequality $2x + 1 > 3x - 4$?

    $x < 5$. Rearrange: $2x - 3x > -4 - 1$, so $-x > -5$, thus $x < 5$.

81. What happens to an inequality when you multiply both sides by a positive number $k>0$?

    Keep the inequality direction the same. Multiplying by a positive number preserves the inequality direction.

82. Which value satisfies $2x-3\le 5$: $x=3$ or $x=5$?

    $x=3$. $2(3)-3=3\le 5$ is true; $2(5)-3=7>5$ is false.

83. When graphing $y\ge -x+2$, should the boundary line be solid or dashed?

    Solid. Solid line includes points on the boundary ($\ge$ includes equality).

84. What is the solution to the absolute value inequality $|x-2|\le 3$?

    $-1\le x\le 5$. Distance from 2 is at most 3.

85. What is the solution to the inequality $0.5x-1<2$?

    $x<6$. Add 1, then divide by 0.5.

86. What is the graph symbol for $x < a$ on a number line?

    Open circle at $a$; shade to the left. Open circle excludes the endpoint; shade shows valid values.

87. What is the solution set of $-1\le x<4$ written in interval notation?

    $[-1,4)$. Bracket at $-1$, parenthesis at $4$ match the inequality symbols.

88. What is the solution to the inequality $-2x>8$?

    $x<-4$. Divide by $-2$ and flip the inequality sign.

89. What is the solution set of $5-2x<1$?

    $x>2$. Subtract 5, divide by -2, and flip the sign.

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

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

91. Which values satisfy either inequality $x < -2$ or $x \ge 1$?

    $(-\infty,-2) \cup [1,\infty)$. Union includes all values satisfying at least one inequality.

92. What is the solution to the inequality $5-3x\ge 2$?

    $x\le 1$. Subtract 5, divide by $-3$, and flip the sign.

93. What is the solution to the inequality $3x\ge 18$?

    $x\ge 6$. Divide both sides by 3: $18÷3=6$.

94. What is the solution to the inequality $7\ge 2x-1$?

    $x\le 4$. Add $1$, then divide by $2$ to isolate $x$.

95. What is the solution to the inequality $x+5<12$?

    $x<7$. Subtract 5 from both sides: $x < 12 - 5$.

96. What is the solution to the inequality $3x \le 18$?

    $x \le 6$. Divide both sides by 3: $x \le \frac{18}{3}$.

97. What is the solution to the inequality $5 - x \le 2$?

    $x \ge 3$. Rearrange: $-x \le 2 - 5 = -3$, so $x \ge 3$.

98. What interval notation matches the inequality $x \ge 3$?

    $[3,\infty)$. Square bracket means $3$ is included; infinity always uses parenthesis.

99. What is the solution set of $x<3$ written in interval notation?

    $(-\infty,3)$. Parenthesis shows $3$ is not included; extends left to negative infinity.

100. What is the solution to the inequality $\frac{x-1}{2} < 5$?

     $x < 11$. Multiply by 2, then add 1: $x - 1 < 10$, so $x < 11$.

101. What line style is used for the boundary when graphing $y\le -x+3$: solid or dashed?

     Solid boundary line. Solid line includes points on the boundary ($\le$ includes equality).

102. What is the solution to the inequality $-3(x+1) \le 6$?

     $x \ge -3$. Distribute: $-3x - 3 \le 6$, then $-3x \le 9$, so $x \ge -3$.

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

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

104. What is the solution to the inequality $5-2(x-3)\ge 1$?

     $x\le 5$. Distribute, combine like terms, then solve for $x$.

105. What is the interval notation for $x>a$?

     $(a,\infty)$. Parenthesis excludes $a$; extends to positive infinity.

106. Which test point is most convenient for checking a half-plane inequality when possible?

     The point $(0,0)$. Origin is easy to substitute unless it's on the boundary line.

107. Which point is a solution to $y\ge x+1$: $(0,0)$ or $(0,2)$?

     $(0,2)$. Test: $2 \ge 0+1$ is true; $(0,0)$ gives $0 \ge 1$ which is false.

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

     $x<14$. Multiply by 3, then add 2: $x - 2 < 12$.

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

     $[-2,\infty)$. Bracket includes $-2$; extends right to positive infinity.

110. What is the solution set of $3x-5<7$?

     $x<4$. Add 5 to get $3x<12$, then divide by 3.

111. What inequality symbol results after dividing both sides by a negative number?

     Reverse the inequality: $\le$ becomes $\ge$ and vice versa. Negative operations flip inequality signs to maintain truth.

112. What is the interval notation for $x\le a$?

     $(-\infty,a]$. Extends from negative infinity; bracket includes $a$.

113. Identify the solution to $\frac{x-1}{-2}\le 3$.

     $x\ge -5$. Multiply by $-2$ (flip sign), then add $1$.

114. Which values satisfy $x<1$ or $x\ge 3$ in interval notation?

     $(-\infty,1)\cup[3,\infty)$. Union symbol combines two separate intervals.

115. Find the integer solutions to $-2<x\le 3$.

     $\{-1,0,1,2,3\}$. Integers greater than $-2$ and at most $3$.

116. What inequality symbol represents "at least" in a statement like "$x$ is at least $5$"?

     $x \ge 5$. "At least" means the value can equal or exceed the given number.

117. What is the interval notation for $x<a$?

     $(-\infty,a)$. Extends from negative infinity; parenthesis excludes $a$.

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

     Reverse the inequality sign (for example, $<$ becomes $>$). Multiplying/dividing by negatives reverses the inequality relationship.

119. What is the interval notation for $-1<x\le 4$?

     $(-1,4]$. Parenthesis excludes -1; bracket includes 4.

120. What is the solution set for $x < a$ on a number line: open or closed circle at $a$?

     Open circle at $a$. Open circle means the endpoint is not included in the solution.

121. What is the interval notation for the solution set of $x>3$?

     $(3,\infty)$. Parenthesis excludes 3; infinity always uses parenthesis.

122. Identify the correct graph endpoint for $x \ge 2$: open circle or closed circle at $2$?

     Closed circle at $2$. Closed circle includes the endpoint value in the solution set.

123. What is the solution to the inequality $-2x > 8$?

     $x < -4$. Divide by $-2$ and flip the sign: $x < \frac{8}{-2}$.

124. What inequality symbol represents "at most" in a statement such as "$x$ is at most $5$"?

     $x \le 5$. "At most" means the value cannot exceed the given number.

125. Identify the solution set in interval notation for $-4x\le 12$.

     $[-3,\infty)$. Divide by $-4$ and flip: $x \ge -3$.

126. What inequality symbol represents "at least" in a statement such as "$x$ is at least $5$"?

     $x \ge 5$. "At least" means the value can equal or exceed the given number.

127. What is the solution set of $-3<\frac{x+1}{2}\le 2$?

     $-7<x\le 3$. Multiply all parts by 2, then subtract 1.

128. What is the solution to the inequality $-2x+1\ge 9$?

     $x\le -4$. Subtract 1, divide by -2 (flipping the sign).

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

     $-2<x\le 2$. Subtract 3 throughout, then divide by 2.

130. What is the solution to the inequality $-2(x+1)<6$?

     $x>-4$. Distribute $-2$, then add $2$ and divide by $-2$ (flip sign).

131. Identify the compound inequality equivalent to $x\in(-2,4]$.

     $-2<x\le 4$. Parenthesis means $>$, bracket means $\le$.

132. What is the solution to the inequality $x + 5 < 12$?

     $x < 7$. Subtract 5 from both sides: $x < 12 - 5$.

133. What is the solution set meaning of $x\le -2$ on a number line?

     All real numbers less than or equal to $-2$. Closed circle at -2, shading extends leftward infinitely.

134. Which number line endpoint is used for $x>2$: open circle or closed circle?

     Open circle at $2$. Open circles exclude the endpoint value.

135. What is the solution to $5-\frac{x}{2}<1$?

     $x>8$. Subtract 5: $-\frac{x}{2} < -4$, multiply by -2 and flip.

136. What is the solution to the compound inequality $3\le x+2<9$?

     $1\le x<7$. Subtract 2 from all parts of the inequality.

137. What inequality symbol represents "at most" in a word problem?

     $\le$. "At most" means less than or equal to.

138. Identify the solution set in interval notation for $\frac{x}{3}-2>1$.

     $(9,\infty)$. Add 2, multiply by 3: $x > 9$.

139. What is the solution set of $-4x\le 12$?

     $x\ge -3$. Divide by -4 and flip the inequality sign.

140. What is the interval notation for $x<-1$?

     $(-\infty,-1)$. Parenthesis excludes -1; negative infinity uses parenthesis.

141. What inequality symbol represents "at least" in a word problem?

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

142. What is the interval notation for the solution set of $x \le -2$?

     $(-\infty, -2]$. Closed bracket includes -2; negative infinity always uses parenthesis.

143. What is the solution to the inequality $\frac{x-2}{-5}\le 3$?

     $x\ge -13$. Multiply by -5 (flip sign), then add 2.

144. What is the solution to the compound inequality $x\le -1$ or $x>2$ in interval notation?

     $(-\infty,-1]\cup(2,\infty)$. Union symbol combines two separate intervals.

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

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

146. Identify the solution to $-3x+2\ge 11$.

     $x\le -3$. Subtract $2$, divide by $-3$, and flip the inequality.

147. Which region is shaded for $y>2x+1$: above or below the boundary line?

     Above the boundary line. Greater than means shade where $y$-values exceed the line.

148. Solve the inequality $\frac{x}{4}-3\ge 2$.

     $x\ge 20$. Add 3, multiply by 4 to isolate $x$.

149. What is the solution set in interval notation for $-3 < x \le 3$?

     $(-3, 3]$. Parenthesis excludes -3; bracket includes 3.

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

     $x> -13$. Multiply by -5 (flipping sign), then add 2 to both sides.

151. What happens to an inequality when you add or subtract the same number on both sides?

     The solution set is unchanged; the inequality direction stays the same. Addition and subtraction preserve inequality relationships.

152. Identify the solution set in interval notation for $2x+1\ge 9$.

     $[4,\infty)$. Solve: $2x \ge 8$, so $x \ge 4$.

153. What change must you make to an inequality when you multiply or divide both sides by a negative number?

     Reverse the inequality sign (e.g., $<$ becomes $>$). Multiplying/dividing by negatives flips the inequality direction.

154. What is the solution to the inequality $-2x > 10$?

     $x < -5$. Divide by -2 and flip the inequality sign.

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

     $(-\infty,-2]$. Closed bracket at -2 includes the endpoint; extends left to negative infinity.

156. Which inequality symbol makes the statement true: If $a<b$, then $-a\ \square\ -b$?

     $>$. Negating reverses the inequality: $-a > -b$.

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

     $-2\le x<2$. Subtract 3 throughout: $-4 \le 2x < 4$, divide by 2.

158. What is the solution to $2(x-3)\ge 4x+2$?

     $x\le -4$. Expand: $2x-6 \ge 4x+2$, isolate x: $-8 \ge 2x$.

159. What is the solution to the inequality $x+3>2x-1$?

     $x<4$. Subtract $x$ and add 1 to both sides.

160. What happens to an inequality when you add the same number $c$ to both sides?

     The solution set is unchanged. Adding or subtracting the same value preserves the inequality relationship.

161. What is the interval notation for the solution set of $x\le -2$?

     $(-\infty,-2]$. Bracket includes -2; negative infinity always uses parenthesis.

162. Which region is shaded for $y>mx+b$: above the line or below the line?

     Above the line. Greater than means shade where y-values exceed the line.

163. What is the solution to $3x-5\le 7$?

     $x\le 4$. Add 5: $3x \le 12$, then divide by 3.

164. What is the interval notation for the compound inequality $-1<x\le 2$?

     $(-1,2]$. Parenthesis at -1 (excluded), bracket at 2 (included).

165. Identify the solution: Solve $4x+1 > 2x+9$.

     $x > 4$. Subtract $2x$ and 1: $2x > 8$, then divide by 2.

166. What is the solution to the inequality $2\le 3x-1\le 8$?

     $1\le x\le 3$. Add 1 throughout, then divide by 3.

167. What is the solution set of $2<x\le 6$ written in interval notation?

     $(2,6]$. Parenthesis at $2$, bracket at $6$ match the inequality symbols.

168. What happens to an inequality sign when you add or subtract the same number on both sides?

     The inequality direction stays the same. Addition/subtraction preserves the inequality relationship.

169. Which inequality matches the statement "$x$ is at most $5$"?

     $x\le 5$. "At most" means less than or equal to.

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

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

171. Which number line endpoint is used for $x\le 2$: open circle or closed circle?

     Closed circle at $2$. Closed circles include the endpoint value.

172. Identify the solution set in interval notation for $3x-5<7$.

     $(-\infty,4)$. Solve: $3x < 12$, so $x < 4$.

173. What is the solution to the inequality $2x-7\le 9$?

     $x\le 8$. Add 7 to both sides, then divide by 2.

174. What is the solution to the absolute value inequality $|x+1|>4$?

     $x<-5$ or $x>3$. Distance from -1 exceeds 4.

175. What is the solution to the inequality $2(x - 3) < 10$?

     $x < 8$. Distribute, then add 6 and divide by 2.

176. Identify the solution: Solve $5-\frac{x}{2} < 1$.

     $x > 8$. Subtract 5: $-\frac{x}{2} < -4$, multiply by -2 and flip.

177. What rule determines whether an inequality sign reverses when you multiply or divide both sides?

     Reverse the sign only when multiplying or dividing by a negative number. Multiplying/dividing by negatives flips the inequality direction.

178. What does a closed circle at $x=-1$ mean on a number line solution graph?

     $x=-1$ is included (uses $\le$ or $\ge$). Closed circle indicates inclusive inequality.

179. Which compound inequality represents all $x$ less than $-2$ or greater than $4$?

     $x<-2$ or $x>4$. "Or" creates two separate regions on the number line.

180. Identify the solution: Solve $-3(x-2) \ge 12$.

     $x \le -2$. Distribute: $-3x+6 \ge 12$, then solve for $x$.

181. What is the solution to $\frac{x}{4}-3\ge 2$?

     $x\ge 20$. Add 3: $\frac{x}{4} \ge 5$, multiply by 4.

182. What is the interval notation for $x\le -1$?

     $(-\infty,-1]$. Bracket includes -1; negative infinity uses parenthesis.

183. What is the solution to the inequality $5-2(3x-1)\ge 1$?

     $x\le 1$. Distribute, combine like terms, solve for $x$.

184. What is the solution to the compound inequality $-2<x\le 5$?

     All $x$ with $-2<x\le 5$. Values between -2 (exclusive) and 5 (inclusive).

185. Identify the solution to $\frac{x}{4}>-2$.

     $x>-8$. Multiply both sides by $4$ to isolate $x$.

186. What inequality symbol represents "at most" in a statement like "$x$ is at most $5$"?

     $x \le 5$. "At most" means the value cannot exceed the given number.

187. What is the graph symbol for $x > a$ on a number line?

     Open circle at $a$; shade to the right. Open circle excludes the endpoint; shade shows valid values.

188. What is the solution to the inequality $-3(x + 2) \le 6$?

     $x \ge -4$. Distribute, divide by -3, and flip the inequality.

189. Find and correct the error: From $-3x\le 12$ someone concluded $x\le -4$.

     Correct: $x\ge -4$. Dividing by -3 requires flipping the inequality sign.

190. Solve the inequality $7-3(x+2)\ge 1$.

     $x\le 0$. Distribute -3, combine terms, then solve for $x$.

191. What interval notation corresponds to $-2 < x \le 5$?

     $(-2,5]$. Parenthesis excludes -2; square bracket includes 5.

192. What is the solution to $7-2(3x+1)<5x$?

     $x>\frac{5}{11}$. Expand: $7 - 6x - 2 < 5x$; simplify: $5 < 11x$.

193. What is the solution set of $2x+1\ge 9$?

     $x\ge 4$. Subtract 1 to get $2x\ge 8$, then divide by 2.

194. What happens to an inequality when you add or subtract the same number on both sides?

     The inequality direction stays the same. Addition and subtraction preserve inequality relationships.

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

     The inequality sign reverses direction (for example, $<$ becomes $>$). Multiplying/dividing by negatives flips the inequality relationship.

196. What is the interval notation for all real numbers $x$ such that $x\le -2$?

     $(-\infty,-2]$. Bracket includes -2; negative infinity always gets parenthesis.

197. What does an open dot mean on a number line when graphing $x < a$ or $x > a$?

     The endpoint $a$ is not included in the solution set. Open dot shows the boundary value is excluded from the solution.

198. What is the solution to the compound inequality $-2<x\le 4$ in interval notation?

     $(-2,4]$. Open at -2, closed at 4 for the compound inequality.

199. What is the solution to the inequality $\frac{x}{4}\ge -3$?

     $x\ge -12$. Multiply both sides by 4 to isolate x.

200. What is the graph symbol for $x \le a$ on a number line?

     Closed circle at $a$; shade to the left. Closed circle includes the endpoint; shade shows valid values.

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

     The inequality sign reverses direction. This rule prevents inequality direction errors when working with negatives.

202. Which value satisfies $-x>4$: $x=-6$ or $x=6$?

     $x=-6$. $-(-6)=6>4$ is true; $-(6)=-6<4$ is false.

203. What is the solution to the inequality $\frac{x}{4} \ge -3$?

     $x \ge -12$. Multiply both sides by 4: $x \ge -3 \times 4$.

204. Does the point $(2,1)$ satisfy $y\le x-1$?

     Yes. Check: $1\le 2-1=1$ is true.

205. What interval notation matches the compound inequality $-3<x\le 2$?

     $(-3,2]$. Parenthesis excludes -3, bracket includes 2.

206. What is the solution to the inequality $3(x-2)\le 12$?

     $x\le 6$. Distribute $3$, add $6$, then divide by $3$.

207. What is the solution to the inequality $-\frac{1}{2}x\le 3$?

     $x\ge -6$. Multiply by $-2$ and flip the inequality sign.

208. What inequality corresponds to the interval notation $(1,4]$?

     $1<x\le 4$. Parenthesis means $>$; bracket means $\le$.

209. What is the interval notation for $x>5$?

     $(5,\infty)$. Parenthesis excludes 5; infinity always uses parenthesis.

210. What does an open circle at $x=a$ mean on a number line graph of an inequality?

     $a$ is not included (inequality is $<$ or $>$). Open circles represent strict inequalities without equality.

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

     Reverse the inequality sign (for example, $<$ becomes $>$). Multiplying/dividing by negatives flips the direction of inequalities.

212. What interval notation matches the compound inequality $-2\le x<4$?

     $[-2,4)$. Bracket at $-2$ (included), parenthesis at $4$ (excluded).

213. What is the solution set of $4\le 2x+6<10$?

     $-1\le x<2$. Subtract 6 from all parts, then divide by 2.

214. What is the solution to the inequality $2x+5<11$?

     $x<3$. Subtract $5$, then divide by $2$ to isolate $x$.

215. Which graph endpoint is used for $x>a$: open circle or closed circle at $a$?

     Open circle at $a$. Open circle excludes the endpoint value.

216. What is the solution set of $-2(x-3)\ge 8$?

     $x\le -1$. Distribute, add 2x and subtract 6, then divide by -2.

217. What does the word "at least" translate to in an inequality?

     $\ge$. "At least" means the minimum value, so greater than or equal to.

218. What is the solution to $5-2x\ge 1+x$?

     $x\le \frac{4}{3}$. Combine like terms: $4 \ge 3x$, then divide by 3.

219. Solve the inequality $5-\frac{x}{2}<1$.

     $x>8$. Subtract 5, multiply by -2, and flip the inequality.

220. What is the solution to the inequality $\frac{x}{4}-2\le 1$?

     $x\le 12$. Add $2$, then multiply by $4$ to isolate $x$.

221. Identify the correct inequality symbol for the phrase: "at least $7$".

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

222. Which inequality sign makes $x=4$ a solution: $x\,\_\,4$ if the set is $\{4,5,6,\dots\}$?

     $x\ge 4$. The set includes 4 and all greater values.

223. What does a closed dot mean on a number line when graphing $x \le a$ or $x \ge a$?

     The endpoint $a$ is included in the solution set. Closed dot shows the boundary value is part of the solution.

224. What is the solution to the compound inequality $2<3x+2\le 11$?

     $0<x\le 3$. Subtract 2, divide by 3 throughout.

225. What is the inequality form for the interval $[-1, 4)$?

     $-1 \le x < 4$. Square bracket means inclusive; parenthesis means exclusive.

226. Identify the correct graph endpoint type for $x>1$: open circle or closed circle at $1$?

     Open circle at $1$. Open circle excludes the endpoint value from the solution.

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

     $-1<x\le 3$. Subtract $1$ from all parts, then divide by $2$.

228. What is the solution to the absolute value inequality $|x| < 3$?

     $-3 < x < 3$. $|x| < 3$ means $x$ is between $-3$ and $3$.

229. Which region is shaded for $y\le mx+b$: above the line or below the line?

     Below the line. Less than or equal means shade where y-values are at or below.

230. Identify the correct inequality symbol for the phrase: "no more than $12$".

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

231. Identify the correct graph endpoint for $x < 2$: open circle or closed circle at $2$?

     Open circle at $2$. Open circle excludes the endpoint value from the solution set.

232. What is the solution to the inequality $\frac{x}{4}>-3$?

     $x>-12$. Multiply both sides by 4.

233. Identify the correct graph endpoint for $x\ge 5$: open circle or closed circle at $5$?

     Closed circle at $5$. Closed circle includes the endpoint value in the solution set.

234. Identify the solution to $2x+3<11$.

     $x<4$. Subtract $3$, then divide by $2$ to isolate $x$.

235. What interval notation represents the inequality $x\ge -2$?

     $[-2,\infty)$. Square bracket shows $-2$ is included; $\infty$ gets parenthesis.

236. What does an open circle at $x=2$ mean on a number line solution graph?

     $x=2$ is not included (uses $<$ or $>$). Open circle indicates strict inequality.

237. Identify the graph boundary type for $y\le -x+3$: solid line or dashed line?

     Solid line. Non-strict inequalities ($\le$ or $\ge$) use solid boundaries.

238. What is the solution to the inequality $3x-5<7$?

     $x<4$. Add 5 to get $3x<12$, then divide by 3.

239. What is the solution to the inequality $x + 7 < 12$?

     $x < 5$. Subtract 7 from both sides: $x < 12 - 7$.

240. What is the solution set for $x \le a$ on a number line: open or closed circle at $a$?

     Closed circle at $a$. Closed circle includes the endpoint in the solution set.

241. Identify the solution set in interval notation for $-\frac{1}{2}x<3$.

     $(-6,\infty)$. Multiply by $-2$ and flip: $x > -6$.

242. What is the interval notation for all real numbers $x$ such that $x>3$?

     $(3,\infty)$. Parenthesis means 3 is not included; infinity always gets parenthesis.

243. Which compound inequality represents all $x$ between $-3$ and $5$, inclusive of both endpoints?

     $-3\le x\le 5$. "Between" with "inclusive" means $\le$ on both sides.

244. What happens to an inequality when you add or subtract the same number on both sides?

     The inequality sign stays the same. Addition and subtraction preserve inequality relationships.

245. What is the solution to the inequality $7 - 2x < 3x + 2$?

     $x > 1$. Collect like terms: $5 > 5x$, then divide by 5.

246. Which endpoint symbol is used on a number line for $x\le a$ or $x\ge a$: open or closed?

     Closed circle. Closed circle indicates the endpoint is included.

247. When graphing $y<\frac{1}{2}x+1$, should the boundary line be solid or dashed?

     Dashed. Dashed line excludes boundary points (strict inequality).

248. What is the boundary line equation used when graphing $y>2x-1$?

     $y=2x-1$. The boundary line is the same equation with equality.

249. What is the solution set of an inequality in one variable?

     All values of the variable that make the inequality true. The solution includes all values satisfying the inequality.

250. What inequality matches the interval notation $(-\infty,5]$?

     $x\le 5$. Square bracket means include 5.

251. What line style is used for the boundary when graphing $y>\frac{1}{2}x+2$: solid or dashed?

     Dashed boundary line. Dashed line excludes boundary points (strict inequality).

252. What is the solution to the inequality $3x \le 18$?

     $x \le 6$. Divide both sides by 3: $x \le \frac{18}{3}$.

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