ACT Math Flashcards: Inequalities And Absolute Value

What this deck covers

This deck focuses on Inequalities And Absolute Value, giving you a quick way to review the definitions, rules, and examples that matter most for ACT 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 solution to $5-2x<1$?

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

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

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

Flashcard 3: What is the numerical distance between $-2$ and $5$ on the number line?

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

Flashcard 4: What is the distance between $-2$ and $5$ on the number line written with absolute value?

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

Flashcard 5: What happens to an inequality when you multiply both sides by a negative number?

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

Flashcard 6: What happens to an inequality when you add the same number to both sides?

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

Flashcard 7: Solve the inequality: $5 - 2x < 1$.

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

Flashcard 8: What is $|5|$?

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

Flashcard 9: What inequality corresponds to the interval $(-1,2]$?

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

Flashcard 10: What is the solution to $x^2 \leq 16$?

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

Flashcard 11: What happens to an inequality when you add the same number to both sides?

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

Flashcard 12: What is the solution to $|x-3|=0$?

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

Flashcard 13: What is the solution to $|x-4|\ge 0$?

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

Flashcard 14: What is $|0|$?

Answer:

1. Absolute value of zero is always zero.

Flashcard 15: Solve for $x$: $|x - 3| = 5$.

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

Flashcard 16: Solve the inequality: $2x + 3 > 7$.

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

Flashcard 17: What is the solution set for $|x| < 4$?

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

Flashcard 18: What does the inequality symbol $\leq$ mean?

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

Flashcard 19: Solve for $x$: $3x - 4 \neq 5$.

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

Flashcard 20: What is $|0|$?

Answer: $0$. Absolute value of zero equals zero.

Flashcard 21: What is the numerical distance between $-2$ and $5$ on the number line?

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

Flashcard 22: What is the solution set of $x<5$ written in interval notation?

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

Flashcard 23: What compound inequality corresponds to $(-\infty,1)\cup(4,\infty)$?

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

Flashcard 24: What inequality corresponds to the interval $[-3,7)$?

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

Flashcard 25: What is the absolute value of 0?

Answer:

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

Flashcard 26: What is the property of $|a + b|$?

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

Flashcard 27: Find $x$ if $|3x + 4| = 7$.

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

Flashcard 28: What is the solution to $|x| \geq 6$?

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

Flashcard 29: What is the solution to $|x+1|\le 3$?

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

Flashcard 30: Solve the inequality: $4x - 5 \geq 3$.

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

Flashcard 31: What is the solution set of $x\ge -2$ written in interval notation?

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

Flashcard 32: What is the solution to $|x|\ge 6$?

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

Flashcard 33: Solve for $x$: $|x + 7| > 4$.

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

Flashcard 34: What is the solution to $|x|>2$?

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

Flashcard 35: What compound inequality corresponds to the interval $[2,5]$?

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

Flashcard 36: What is the absolute value of -7?

Answer:

1. Absolute value converts negative numbers to positive.

Flashcard 37: Find $x$ if $|2x - 1| = 3$.

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

Flashcard 38: What is $|0|$?

Answer:

1. Absolute value of zero is always zero.

Flashcard 39: What is $|5|$?

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

Flashcard 40: Solve the inequality: $2x + 3 > 7$.

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

Flashcard 41: What is the solution set of $x<5$ written in interval notation?

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

Flashcard 42: What is the solution to $|x-4|<0$?

Answer: No solution. Absolute value cannot be negative.

Flashcard 43: What is the solution to $|x-4|<0$?

Answer: No solution. Absolute value cannot be negative.

Flashcard 44: What inequality corresponds to the interval $[-3,7)$?

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

Flashcard 45: What is the solution to $|x|<4$?

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

Flashcard 46: What is the solution to $2x+1\le 9$?

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

Flashcard 47: What is the solution to $x^2 > 9$?

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

Flashcard 48: Solve for $x$: $|x + 2| \geq 3$.

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

Flashcard 49: What is $|-7|$?

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

Flashcard 50: What is the solution set of $x\le 4$ written in interval notation?

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

Flashcard 51: Solve for $x$: $3x - 4 \neq 5$.

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

Flashcard 52: Solve for $x$: $2 < 3x + 1 \leq 8$.

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

Flashcard 53: What is the distance between $-2$ and $5$ on the number line written with absolute value?

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

Flashcard 54: What is the result of $|-9|$?

Answer:

1. Absolute value converts negative to positive.

Flashcard 55: What is the solution to $|3x+6|<9$?

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

Flashcard 56: What is the solution to $3x-5>7$?

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

Flashcard 57: Solve for $x$: $2 < 3x + 1 \leq 8$.

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

Flashcard 58: What inequality corresponds to the interval $(-1,2]$?

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

Flashcard 59: What is the solution to $x^2 > 9$?

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

Flashcard 60: What is the solution set of $x\ge -2$ written in interval notation?

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

Flashcard 61: What is the solution to $x\ne 4$ in interval notation?

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

Flashcard 62: What does the inequality symbol $\geq$ mean?

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

Flashcard 63: What is the absolute value of -12?

Answer:

1. Absolute value makes negative numbers positive.

Flashcard 64: What is the solution to $|2x-1|\ge 5$?

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

Flashcard 65: What is the solution to $\frac{x-3}{2}<5$?

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

Flashcard 66: What is the solution to $|x-3|=0$?

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

Flashcard 67: Solve the inequality: $x + 5 \leq 9$.

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

Flashcard 68: What is the inequality for 'at most 10'?

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

Flashcard 69: What is the solution to $|3x+6|<9$?

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

Flashcard 70: Solve for $x$: $|x - 3| = 5$.

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

Flashcard 71: What is the solution to $|x-2|=7$?

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

Flashcard 72: Solve for $x$: $|x - 4| < 2$.

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

Flashcard 73: Find $x$ if $|3x + 4| = 7$.

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

Flashcard 74: What is the solution to the compound inequality $-1\le 2x+3<7$?

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

Flashcard 75: What is the solution to $|x|<4$?

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

Flashcard 76: What is the solution to $|x-2|=7$?

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

Flashcard 77: What is the graphical representation of $|x|$ on a number line?

Answer: Distance from 0. Absolute value measures distance from origin.

Flashcard 78: What is the solution set for $|x| = 0$?

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

Flashcard 79: What is the solution to the compound inequality $3<\frac{x+1}{2}\le 5$?

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

Flashcard 80: Solve for $x$: $|x + 7| > 4$.

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

Flashcard 81: What is $|0|$?

Answer: $0$. Absolute value of zero equals zero.

Flashcard 82: What is the solution to $|x-4|\ge 0$?

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

Flashcard 83: What is the solution to $|x-5|<2$?

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

Flashcard 84: Solve the inequality: $5 - 2x < 1$.

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

Flashcard 85: What is the inequality for 'at least 15'?

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

Flashcard 86: What is the solution to $\frac{x-3}{2}<5$?

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

Flashcard 87: What is the definition of absolute value $|a|$ for real $a$?

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

Flashcard 88: Solve the inequality: $-3x + 4 > 1$.

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

Flashcard 89: Describe the inequality $x > 5$.

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

Flashcard 90: What is the solution to $x^2 \leq 16$?

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

Flashcard 91: What is $|-7|$?

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

Flashcard 92: What does the inequality symbol $>$ mean?

Answer: Greater than. Symbol for strict greater than comparison.

Flashcard 93: What is the solution set of $x\le 4$ written in interval notation?

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

Flashcard 94: What is the solution to $|x-1|\le 0$?

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

Flashcard 95: Solve for $x$: $x^2 - 4x < 0$.

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

Flashcard 96: What is the solution to $-4 < x < 2$?

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

Flashcard 97: What is the solution to $|x|\le 3$?

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

Flashcard 98: What is the solution to $|2x+3|=1$?

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

Flashcard 99: Solve the inequality: $-3x + 4 > 1$.

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

Flashcard 100: What is the solution set for $|x| < 4$?

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