PSAT Math Flashcards: Linear Inequalities

What this deck covers

This deck focuses on Linear Inequalities, giving you a quick way to review the definitions, rules, and examples that matter most for PSAT Math.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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