Linear Equations and Inequalities - GRE Quantitative Reasoning

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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