GRE Quantitative Flashcards: Linear Equations Inequalities

What this deck covers

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

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: Identify the solution set for $|x-2|\le 5$.

Answer: $-3\le x\le 7$. Rewriting $|x-2| \le 5$ as $-5 \le x-2 \le 5$ and adding 2 to all parts solves the inequality.

Flashcard 2: Identify the $y$-intercept of the line $3x+2y=8$.

Answer: $4$. Rewriting $3x + 2y = 8$ in slope-intercept form reveals the y-intercept as 4.

Flashcard 3: Identify the equation of the line with slope $2$ and $y$-intercept $-3$.

Answer: $y=2x-3$. Substituting slope 2 and y-intercept $-3$ into the slope-intercept form gives the equation.

Flashcard 4: What happens to an inequality when you multiply or divide both sides by a negative number?

Answer: Reverse the inequality sign. Multiplying or dividing an inequality by a negative number reverses the direction to preserve the inequality's validity.

Flashcard 5: What is the condition for two nonvertical lines to be parallel in terms of slopes $m_1$ and $m_2$?

Answer: $m_1=m_2$. Nonvertical parallel lines share the same slope, ensuring they maintain a constant distance without intersecting.

Flashcard 6: What is the slope of the line given by $Ax+By=C$ (with $B\ne 0$)?

Answer: $m=-\frac{A}{B}$. Rearranging the standard form to slope-intercept yields the slope as the negative ratio of coefficients $A$ to $B$.

Flashcard 7: Identify the solution for $\frac{x}{4}+3=5$.

Answer: $x=8$. Subtracting 3 from both sides and multiplying by 4 solves for $x$.

Flashcard 8: What is the equation of a horizontal line passing through $y=b$?

Answer: $y=b$. A horizontal line includes all points with the constant y-coordinate $b$, yielding a slope of zero.

Flashcard 9: What inequality is equivalent to $|x|<a$ for $a>0$?

Answer: $-a<x<a$. The absolute value inequality $|x| < a$ means $x$ lies strictly between $-a$ and $a$.

Flashcard 10: What is the slope-intercept form of a line?

Answer: $y=mx+b$. This form directly incorporates the slope $m$ and y-intercept $b$ to define the linear equation.

Flashcard 11: What inequality is equivalent to $|x|\le a$ for $a\ge 0$?

Answer: $-a\le x\le a$. The absolute value inequality $|x| \le a$ includes all $x$ from $-a$ to $a$ inclusive.

Flashcard 12: What is the point-slope form of a line through $(x_1,y_1)$ with slope $m$?

Answer: $y-y_1=m(x-x_1)$. This form utilizes a known point $(x_1, y_1)$ and the slope $m$ to express the line's equation.

Flashcard 13: Identify the equation of the line through $(1,4)$ with slope $-2$.

Answer: $y=-2x+6$. Using point-slope form with point $(1,4)$ and slope $-2$, then simplifying, yields the equation.

Flashcard 14: Identify the slope of the line through $(2,5)$ and $(6,1)$.

Answer: $m=-1$. Applying the slope formula to the points $(2,5)$ and $(6,1)$ gives the value.

Flashcard 15: What compound inequality is equivalent to $|x|>a$ for $a>0$?

Answer: $x<-a$ or $x>a$. The absolute value inequality $|x| > a$ indicates $x$ is either less than $-a$ or greater than $a$.

Flashcard 16: What compound inequality is equivalent to $|x|\ge a$ for $a\ge 0$?

Answer: $x\le -a$ or $x\ge a$. The absolute value inequality $|x| \ge a$ includes $x$ at or below $-a$ or at or above $a$.

Flashcard 17: Identify the solution set for $5x+1\le 16$.

Answer: $x\le 3$. Subtracting 1 and dividing by 5, with the inequality direction unchanged, yields the solution set.

Flashcard 18: What is the standard form of a linear equation in two variables?

Answer: $Ax+By=C$. This general form represents linear equations where $A$, $B$, and $C$ are constants with $A$ and $B$ not both zero.

Flashcard 19: Identify the solution set for $-2x>6$.

Answer: $x<-3$. Dividing by $-2$ reverses the inequality, resulting in the solution set.

Flashcard 20: What is the equation of a vertical line passing through $x=a$?

Answer: $x=a$. A vertical line consists of all points sharing the fixed x-coordinate $a$, resulting in an undefined slope.

Flashcard 21: Identify the solution set for $3<2x+1\le 9$.

Answer: $1<x\le 4$. Subtracting 1 from all parts and dividing by 2 solves the compound inequality.

Flashcard 22: Identify the solution for $3x-5=16$.

Answer: $x=7$. Adding 5 to both sides and dividing by 3 isolates $x$ to find the solution.

Flashcard 23: What is the slope between points $(x_1,y_1)$ and $(x_2,y_2)$ with $x_2\ne x_1$?

Answer: $m=\frac{y_2-y_1}{x_2-x_1}$. The slope formula computes the ratio of the change in $y$-coordinates to the change in $x$-coordinates between two points.

Flashcard 24: Identify the solution for $2(x-3)=10$.

Answer: $x=8$. Dividing both sides by 2 and adding 3 isolates $x$.

Flashcard 25: What is the condition for two nonvertical lines to be perpendicular in terms of slopes $m_1$ and $m_2$?

Answer: $m_1m_2=-1$. Nonvertical perpendicular lines have slopes that are negative reciprocals, with their product equaling $-1$.