Your FREE Praxis Algebra I (5162) Flashcards 2026 – 200+ Cards

A letter or symbol that represents an unknown or changing quantity. In $3x + 5$, the letter $x$ is the variable.

A combination of numbers, variables, and operations with no equals sign, such as $4x^2 - 7x + 2$.

The numerical factor multiplying a variable. In $6x$ the coefficient is $6$; a lone $x$ has coefficient $1$.

A term with no variable, so its value never changes. In $2x + 9$ the constant term is $9$.

Terms with the same variables raised to the same powers, so they can be combined. $3x$ and $5x$ are like terms; $3x$ and $3x^2$ are not.

Adding or subtracting the coefficients of like terms. For example, $7x + 2x = 9x$ and $5x^2 - x^2 = 4x^2$.

Multiplying a sum by distributing the factor to each term: $a(b + c) = ab + ac$. So $3(x + 4) = 3x + 12$.

Substituting given numbers for the variables and simplifying. Evaluating $2x + 1$ at $x = 5$ gives $11$.

An equation where the variable appears only to the first power, such as $3x - 7 = 11$. Its graph on a number line is a single point.

A value of the variable that makes the equation true. For $3x - 7 = 11$ the solution is $x = 6$.

An equation solved with a single inverse operation. For $x + 8 = 13$, subtract $8$ to get $x = 5$.

An equation requiring two inverse operations. For $2x + 3 = 11$, subtract $3$ then divide by $2$ to get $x = 4$.

Collect variable terms on one side and constants on the other. For $5x = 2x + 9$, subtract $2x$: $3x = 9$, so $x = 3$.

Operations that undo each other, used to isolate a variable: addition and subtraction, or multiplication and division.

Solving a formula for a different variable using inverse operations. Solving $d = rt$ for $t$ gives $t = \dfrac{d}{r}$.

A statement comparing two expressions with $<$, $>$, $\le$, or $\ge$, such as $2x + 1 \le 9$.

When you multiply or divide both sides of an inequality by a negative number, reverse the sign. From $-2x < 6$ you get $x > -3$.

Two inequalities joined by "and" or "or," such as $-3 < x \le 5$, which means $x$ lies between $-3$ and $5$.

Use an open circle for $<$ or $>$ and a closed circle for $\le$ or $\ge$, then shade toward the solution set.

The region of the coordinate plane representing a two-variable linear inequality, bounded by a solid line for $\le, \ge$ or a dashed line for $<, >$.

Two or more linear equations considered together; the solution is the point that satisfies all of them at once.

Solve one equation for a variable, then substitute that expression into the other equation to solve a system.

Add or subtract multiples of the equations to cancel one variable. Adding $x + y = 5$ and $x - y = 1$ gives $2x = 6$.

Plot both lines; their intersection point is the solution. Parallel lines give no solution; identical lines give infinitely many.

Occurs when two lines are parallel (same slope, different intercepts), so they never meet — the system is inconsistent.

Occurs when the two equations represent the same line, so every point on it is a solution — the system is dependent.

A small raised number showing how many times the base is multiplied by itself. In $2^4$ the base $2$ is used as a factor four times, giving $16$.

When multiplying powers with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$. So $x^2 \cdot x^5 = x^7$.

When dividing powers with the same base, subtract the exponents: $\dfrac{a^m}{a^n} = a^{m-n}$. So $\dfrac{x^7}{x^3} = x^4$.

When raising a power to a power, multiply the exponents: $(a^m)^n = a^{mn}$. So $(x^3)^4 = x^{12}$.

A product raised to a power distributes to each factor: $(ab)^n = a^n b^n$. So $(2x)^3 = 8x^3$.

Any nonzero base raised to the zero power equals $1$: $a^0 = 1$. For example, $7^0 = 1$.

A negative exponent means the reciprocal: $a^{-n} = \dfrac{1}{a^n}$. So $2^{-3} = \dfrac{1}{8}$.

An expression of one or more terms made of variables with whole-number exponents, such as $3x^2 + 2x - 5$.

The greatest exponent on the variable. The degree of $4x^3 - x + 7$ is $3$.

Combine like terms. $(3x^2 + 2x) + (x^2 - 5x) = 4x^2 - 3x$.

Distribute the minus sign, then combine like terms. $(5x^2 + x) - (2x^2 - 3x) = 3x^2 + 4x$.

Multiply every term of one polynomial by every term of the other, then combine like terms, as with the FOIL method for two binomials.

A way to multiply two binomials: First, Outer, Inner, Last. $(x + 3)(x + 2) = x^2 + 5x + 6$.

Pull the greatest common factor from every term: $6x^2 + 9x = 3x(2x + 3)$.

The largest expression that divides each term evenly. The GCF of $8x^3$ and $12x^2$ is $4x^2$.

Write $x^2 + bx + c$ as a product of two binomials whose constants multiply to $c$ and add to $b$. So $x^2 + 5x + 6 = (x + 2)(x + 3)$.

Factors as conjugate binomials: $a^2 - b^2 = (a + b)(a - b)$. So $x^2 - 9 = (x + 3)(x - 3)$.

A trinomial that factors as a binomial squared: $a^2 + 2ab + b^2 = (a + b)^2$. So $x^2 + 6x + 9 = (x + 3)^2$.

A ratio of two polynomials, such as $\dfrac{x^2 - 1}{x + 1}$, defined wherever the denominator is not zero.

Factor numerator and denominator, then cancel common factors. $\dfrac{x^2 - 1}{x + 1} = \dfrac{(x+1)(x-1)}{x+1} = x - 1$.

An input that makes a denominator zero and is therefore not allowed. In $\dfrac{1}{x - 3}$ the excluded value is $x = 3$.

An expression containing a root symbol, such as $\sqrt{x}$ or $\sqrt{50}$.

Factor out perfect-square factors from under the radical: $\sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}$.

A radical over a product splits into separate radicals: $\sqrt{ab} = \sqrt{a}\,\sqrt{b}$ for nonnegative $a, b$.

The distance of a number from zero, always nonnegative: $|{-7}| = 7$ and $|4| = 4$.

An equation like $|x| = 5$ that splits into two cases, giving $x = 5$ or $x = -5$.

A second-degree equation in the form $ax^2 + bx + c = 0$ with $a \ne 0$, such as $x^2 - 5x + 6 = 0$.

Written as $ax^2 + bx + c = 0$, with all terms on one side and the other side equal to zero.

If a product equals zero, at least one factor is zero. From $(x - 2)(x - 3) = 0$ you get $x = 2$ or $x = 3$.

Factor the quadratic, set each factor to zero, and solve. $x^2 - 5x + 6 = 0$ factors as $(x-2)(x-3)=0$, so $x = 2, 3$.

Solves $ax^2 + bx + c = 0$: $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

The expression $b^2 - 4ac$ under the radical of the quadratic formula. It tells how many real roots a quadratic has.

If $b^2 - 4ac > 0$ there are two real roots; if it equals $0$ there is one; if it is $< 0$ there are no real roots.

Rewriting $ax^2 + bx + c$ in the form $(x - p)^2 = q$ to solve a quadratic or find a vertex.

A value that makes the equation equal zero; the roots of $x^2 - 4 = 0$ are $x = 2$ and $x = -2$.

Expressions that have the same value for every input, such as $2(x + 3)$ and $2x + 6$.

Isolating one chosen variable in an equation containing several letters, treating the rest as constants.

Translating a real-world relationship into an equation, such as writing $C = 25 + 0.10m$ for a cost with a base fee and a per-mile rate.

Substituting a found value back into the original equation to confirm it makes the statement true.

The multiplicative inverse of a nonzero number: the reciprocal of $\dfrac{2}{3}$ is $\dfrac{3}{2}$, and a number times its reciprocal is $1$.

Set the cross products equal and solve. From $\dfrac{x}{4} = \dfrac{3}{6}$, cross-multiplying gives $6x = 12$, so $x = 2$.

If $x^2 = q$ with $q \ge 0$, then $x = \pm\sqrt{q}$. So $x^2 = 16$ gives $x = 4$ or $x = -4$.

A single-term polynomial, such as $5x^3$ or $-7$.

A polynomial with exactly two terms, such as $3x + 2$.

A polynomial with exactly three terms, such as $x^2 + 5x + 6$.

Group a four-term polynomial in pairs, factor each pair, then factor out the shared binomial, as in $x^3 + x^2 + x + 1 = (x+1)(x^2+1)$.

Use the same steps as an equation, but reverse the sign when multiplying or dividing by a negative; the answer is a range of values.

An equation true for every value of the variable, such as $2(x + 1) = 2x + 2$, which has infinitely many solutions.

An equation that reduces to a false statement, such as $x + 1 = x + 2$, which gives $1 = 2$ and has no solution.

For $\dfrac{a}{b} = \dfrac{c}{d}$, the products $ad$ and $bc$ are equal; used to solve proportions.

A rule that assigns exactly one output to each input. Each $x$-value maps to a single $y$-value.

Writing $f(x)$ to name the output for input $x$. For $f(x) = 2x + 1$, $f(3) = 7$.

Substituting an input value for the variable. For $f(x) = x^2 - 1$, $f(4) = 15$.

The set of all allowable input values. For $f(x) = \dfrac{1}{x - 2}$ the domain is all real numbers except $x = 2$.

The set of all output values the function produces. For $f(x) = x^2$ the range is $y \ge 0$.

A graph represents a function if every vertical line crosses it at most once, since each input has one output.

The input of a function, usually $x$, whose value is chosen freely.

The output of a function, usually $y$ or $f(x)$, whose value depends on the input.

A function whose graph is a straight line, written $f(x) = mx + b$, with a constant rate of change $m$.

The steepness of a line, equal to rise over run: $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.

A line that rises from left to right, where $y$ increases as $x$ increases, so $m > 0$.

A line that falls from left to right, where $y$ decreases as $x$ increases, so $m < 0$.

A horizontal line of the form $y = b$, where the output never changes, so $m = 0$.

A vertical line of the form $x = a$; its run is zero, so the slope is undefined and it is not a function.

The point where a graph crosses the $y$-axis, where $x = 0$. For $y = 2x + 5$ the $y$-intercept is $(0, 5)$.

The point where a graph crosses the $x$-axis, where $y = 0$; also called a zero of the function.

A linear equation written $y = mx + b$, where $m$ is the slope and $b$ is the $y$-intercept.

A linear equation written $y - y_1 = m(x - x_1)$, using a known point $(x_1, y_1)$ and the slope $m$.

A linear equation written $Ax + By = C$, with $A, B, C$ constants and $A, B$ not both zero.

Distinct lines with equal slopes that never intersect. Lines $y = 2x + 1$ and $y = 2x - 4$ are parallel.

Lines that meet at a right angle; their slopes are negative reciprocals, so $m_1 \cdot m_2 = -1$.

A function of the form $f(x) = ax^2 + bx + c$ with $a \ne 0$, whose graph is a parabola.

The U-shaped graph of a quadratic function, opening upward when $a > 0$ and downward when $a < 0$.

The highest or lowest point of a parabola, where it changes direction; the $x$-coordinate is $x = -\dfrac{b}{2a}$.

The vertical line $x = -\dfrac{b}{2a}$ that splits a parabola into mirror-image halves through its vertex.

A quadratic written $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.

The $y$-coordinate of the vertex when the parabola opens downward ($a < 0$); the function never exceeds it.

The $y$-coordinate of the vertex when the parabola opens upward ($a > 0$); the function never goes below it.

The input values where the output is zero, i.e. where the graph crosses the $x$-axis; the roots of $f(x) = 0$.

A function of the form $f(x) = a \cdot b^x$ with base $b > 0$, $b \ne 1$, where the output changes by a constant factor.

An exponential function with base $b > 1$, so the quantity increases by a constant percent each step, as in $f(x) = 100(1.05)^x$.

An exponential function with base $0 < b < 1$, so the quantity decreases by a constant percent each step, as in $f(x) = 100(0.5)^x$.

The base of an exponential growth model; a $5\%$ increase gives a growth factor of $1.05$.

The change in output over the change in input on an interval: $\dfrac{f(b) - f(a)}{b - a}$, like a slope between two points.

A rate that stays the same across all intervals, which is the defining feature of a linear function.

A function whose output rises as the input rises, so the graph goes up from left to right on that interval.

A function whose output falls as the input rises, so the graph goes down from left to right on that interval.

Shifting a graph up or down: $g(x) = f(x) + k$ moves it up $k$ units when $k > 0$.

Shifting a graph left or right: $g(x) = f(x + k)$ moves it left $k$ units when $k > 0$.

Multiplying the output scales a graph vertically: $g(x) = k\,f(x)$ stretches it when $k > 1$ and compresses it when $0 < k < 1$.

Negating the output flips a graph vertically: $g(x) = -f(x)$.

A function defined by different rules on different parts of its domain, such as one formula for $x < 0$ and another for $x \ge 0$.

The function $f(x) = |x|$, whose V-shaped graph has its vertex at the origin and is always nonnegative.

A function whose graph is a series of horizontal segments, jumping between constant values, like the greatest integer function.

An ordered list of numbers called terms, such as $2, 5, 8, 11, \dots$, often defined as a function of the term number.

A sequence with a constant difference between consecutive terms, such as $3, 7, 11, 15, \dots$ with common difference $4$.

The fixed amount added to each term of an arithmetic sequence to get the next term. In $5, 8, 11, \dots$ it is $3$.

The $n$th term is $a_n = a_1 + (n - 1)d$. For $a_1 = 3$, $d = 4$, the $20$th term is $3 + 19 \cdot 4 = 79$.

A sequence with a constant ratio between consecutive terms, such as $2, 6, 18, 54, \dots$ with common ratio $3$.

The fixed factor each term of a geometric sequence is multiplied by to get the next. In $4, 8, 16, \dots$ it is $2$.

The $n$th term is $a_n = a_1 \cdot r^{\,n - 1}$. For $a_1 = 2$, $r = 3$, the $4$th term is $2 \cdot 3^3 = 54$.

A rule that defines each term using the previous term, such as $a_n = a_{n-1} + 4$ with $a_1 = 3$.

A rule that gives any term directly from its position $n$, such as $a_n = 3 + 4(n - 1)$.

Reading features such as intercepts, slope, maxima, minima, and intervals of increase or decrease to describe what a function does.

Applying one function to the output of another, written $(f \circ g)(x) = f(g(x))$.

A function that reverses another, swapping inputs and outputs; if $f(2) = 7$ then $f^{-1}(7) = 2$.

Linear models grow by equal differences each step, while exponential models grow by equal factors, so exponential eventually outpaces linear.

Growth modeled by $A(t) = Pe^{rt}$, where $P$ is the initial amount, $r$ the rate, and $e \approx 2.718$.

A relationship where $y = \dfrac{k}{x}$, so the product $xy$ is constant and $y$ decreases as $x$ increases.

A relationship where $y = kx$, so $y$ is a constant multiple of $x$ and the graph passes through the origin.

All rational and irrational numbers, which together make up every point on the number line.

A number that can be written as a fraction $\dfrac{a}{b}$ of integers with $b \ne 0$, such as $\dfrac{3}{4}$, $-2$, or $0.25$.

A number that cannot be written as a fraction of integers; its decimal never ends or repeats, such as $\sqrt{2}$ or $\pi$.

A whole number or its opposite, with no fractional part: $\dots, -2, -1, 0, 1, 2, \dots$.

A set is closed under addition if adding any two of its members stays in the set. The rationals are closed under addition.

The sum of two rational numbers is always rational, since fractions of integers add to another fraction of integers.

The sum of a rational and an irrational number is always irrational, such as $3 + \sqrt{2}$.

Multiplying a nonzero rational by an irrational number always gives an irrational result, such as $2\sqrt{3}$.

Order does not affect a sum or product: $a + b = b + a$ and $ab = ba$.

Grouping does not affect a sum or product: $(a + b) + c = a + (b + c)$ and $(ab)c = a(bc)$.

Evaluate in the order parentheses, exponents, multiplication and division, then addition and subtraction (PEMDAS).

Writing a number as $a \times 10^n$ with $1 \le a < 10$. For example, $4{,}500 = 4.5 \times 10^3$.

The power of ten that best describes a quantity's size; $3 \times 10^6$ is one order of magnitude larger than $3 \times 10^5$.

An exponent written as a fraction that means a root: $a^{1/2} = \sqrt{a}$ and $a^{m/n} = \sqrt[n]{a^m}$.

Using units as factors to convert measurements, multiplying by ratios equal to $1$ so the unwanted units cancel.

A rate with a denominator of $1$, such as $60$ miles per $1$ hour, found by dividing the two quantities.

A comparison of two quantities by division, written $a : b$ or $\dfrac{a}{b}$, such as $3 : 4$.

An equation stating two ratios are equal, such as $\dfrac{3}{4} = \dfrac{9}{12}$.

A ratio out of $100$. For example, $25\% = \dfrac{25}{100} = 0.25$.

Judging whether a result makes sense in context by estimating and checking units before accepting it.

Accuracy is how close a measurement is to the true value; precision is how consistent repeated measurements are with one another.

The arithmetic average: add all values and divide by how many there are. The mean of $4, 6, 8$ is $6$.

The middle value of an ordered data set; with an even count it is the average of the two middle values. It resists outliers.

The value that appears most often in a data set. In $2, 3, 3, 5$ the mode is $3$.

The spread from lowest to highest value: maximum minus minimum. For $4, 7, 12$ the range is $8$.

A single value summarizing the middle of a data set, such as the mean, median, or mode.

A value describing how varied a data set is, such as the range or the standard deviation.

A measure of how far data values typically fall from the mean; a larger value means the data are more spread out.

A data value far from the rest of the set, which can strongly pull the mean but has little effect on the median.

A graph of paired data as points on a coordinate plane, used to reveal the relationship between two variables.

A straight line drawn through scattered data to model the trend and make predictions; also called a trend line.

The strength and direction of a linear relationship between two variables, which may be positive, negative, or near zero.

A pattern in which one variable tends to increase as the other increases, so the points trend upward.

A pattern in which one variable tends to decrease as the other increases, so the points trend downward.

Two variables can be correlated without one causing the other; a relationship alone does not prove cause and effect.

A number from $0$ to $1$ measuring how likely an event is: favorable outcomes divided by total equally likely outcomes.

An event with a single outcome, such as rolling a $4$ on one die, which has probability $\dfrac{1}{6}$.

The set of outcomes where the event does not occur; its probability is $1 - P(A)$.

An event combining two or more simple events with "and" or "or," such as drawing a red card and then a king.

Events where one occurring does not affect the other; their joint probability is the product $P(A) \cdot P(B)$.

Events where the outcome of one changes the probability of the other, as in drawing cards without replacement.

Events that cannot happen at the same time; for them $P(A \text{ or } B) = P(A) + P(B)$.

For any two events, $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

The probability that both independent events occur is $P(A \text{ and } B) = P(A) \cdot P(B)$.

Probability found by reasoning about equally likely outcomes, such as $\dfrac{1}{2}$ for heads on a fair coin.

Probability estimated from observed results: the number of times an event happened divided by the number of trials.

The set of all possible outcomes of an experiment; for one die it is $\{1, 2, 3, 4, 5, 6\}$.

If one choice has $m$ options and another has $n$, the number of combined outcomes is $m \times n$.

An arrangement of items where order matters, such as the number of ways to seat people in a row.

A selection of items where order does not matter, such as choosing a committee from a group.

The product of all positive integers up to $n$, written $n!$. For example, $4! = 4 \cdot 3 \cdot 2 \cdot 1 = 24$.

A bar graph showing the frequency of data grouped into intervals, with no gaps between bars.

A display summarizing data with its minimum, lower quartile, median, upper quartile, and maximum.

A value dividing ordered data into four equal parts; the median is the second quartile.

The spread of the middle half of the data: the third quartile minus the first quartile, $Q_3 - Q_1$.

The number of times a value or category occurs in a data set.

Data sorted into groups or labels, such as colors or brands, rather than measured numerically.

Data made of measured or counted numbers, such as heights or test scores, that can be averaged.

The spacing of values along an axis; a misleading scale can exaggerate or hide differences in data.

The point $(0, 0)$ where the $x$-axis and $y$-axis intersect on the coordinate plane.

The digits in a measurement that carry meaning about its precision, including all certain digits plus one estimated digit.

Finding an approximate value, often by rounding, to check that a calculated answer is reasonable.

On a number line, $|a - b|$ gives the distance between $a$ and $b$, which is never negative.

The number that, multiplied by itself three times, gives the radicand: $\sqrt[3]{27} = 3$ because $3^3 = 27$.