Algebra 1 Midterm Exam Review

Vocabulary practice flashcards reviewing core Algebra 1 midterm concepts, including properties of real numbers, equation forms, inequalities, sequence properties, line relationships, and systems of linear equations.

Natural Numbers

The set of positive integers used for counting, starting from $1$ ($1, 2, 3, \dots$), which contains values such as $22$.

Rational Numbers

The set of real numbers that can be written as a fraction or quotient $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Division Property of Equality

The algebraic property stating that dividing both sides of an equation by the same nonzero value yields an equivalent equation, such as dividing by $-8$ in $\frac{42}{-8} = \frac{-8x}{-8}$.

Trapezoid Area Formula

The geometric formula $A = \frac{1}{2}(b_1 + b_2)h$, which relates area $A$, base lengths $b_1$ and $b_2$, and height $h$; solving for height yields $h = \frac{2A}{b_1 + b_2}$.

Closed Circle (Number Line)

A graphical representation on a number line indicating that the boundary point is included in the solution set, corresponding to the inequality operators $\ge$ or $\le$.

Compound Inequality

A mathematical statement formed by joining two distinct inequalities with the connective word "and" (an intersection) or "or" (a union).

Slope-Intercept Form

The standard linear form $y = mx + b$, where $m$ is the slope of the line and $b$ is the $y$-coordinate of the $y$-intercept.

Positive Slope (Linear Graph)

A line that slants upward from left to right; for example, a line passing through $(0, 1)$ and $(2, 4)$ has a positive slope of $m = \frac{3}{2}$.

Negative Slope (Linear Graph)

A line that slants downward from left to right, indicating that the dependent variable $y$ decreases as the independent variable $x$ increases.

Point-Slope Form

The linear equation form $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ are the coordinates of a known point on the line.

Standard Form of a Linear Equation

A linear equation written in the format $Ax + By = C$, where $A$, $B$, and $C$ are integers, and $A$ and $B$ are not both equal to zero.

Parallel Lines (Slopes)

Two or more lines in the same coordinate plane that never intersect because they share identical slopes ($m_1 = m_2$).

Perpendicular Lines (Slopes)

Two lines that intersect at a $90^\circ$ angle and have slopes that are negative reciprocals of one another, satisfying $m_1 \cdot m_2 = -1$.

Negative Reciprocal

A number that, when multiplied by a given value, produces a product of $-1$; for example, the negative reciprocal of $\frac{3}{8}$ is $-\frac{8}{3}$.

Vertical Stretch Transformation

A function transformation represented by $g(x) = c \cdot f(x)$ where $c > 1$, which multiplies every output ($y$-value) of $f(x)$ by the factor $c$.

Arithmetic Sequence

A sequence of numbers in which each term after the first is obtained by adding a fixed constant value to the preceding term.

Common Difference

The fixed numerical amount $d$ added to each term in an arithmetic sequence to produce the next term (e.g., $d = 4$ in $15, 19, 23, 27, \dots$).

Negative Association

A relationship between two variables on a scatter plot where higher values of one variable generally correspond to lower values of the other variable.

Break-Even Point

The point where total production costs equal total sales revenue, resulting in zero net profit or loss ($\text{Revenue} = \text{Cost}$).

Substitution Method

An algebraic method for solving a system of equations by isolating one variable in an equation and replacing it in the second equation.

Elimination Method

An algebraic method for solving a system of equations by adding or subtracting the equations to cancel out one of the variables.