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

Praxis 5162 Flashcards

Variable

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

Algebraic expression

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

Coefficient

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

Constant term

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

Like terms

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.

Combining like terms

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

Distributive property

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

Evaluating an expression

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

Linear equation in one variable

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.

Solution of an equation

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

One-step equation

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

Two-step equation

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

Variables on both sides

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

Inverse operations

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

Rearranging a formula

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

Linear inequality

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

Flipping the inequality sign

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

Compound inequality

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

Graphing an inequality on a number line

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

Half-plane

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 $<, >$.

System of linear equations

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

Substitution method

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

Elimination method

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

Graphing a system

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

System with no solution

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

System with infinitely many solutions

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

Exponent

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 $16$.

Product rule for exponents

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

Quotient rule for exponents

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

Power of a power rule

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

Power of a product rule

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

Zero exponent

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

Negative exponent

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

Polynomial

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

Degree of a polynomial

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

Adding polynomials

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

Subtracting polynomials

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

Multiplying polynomials

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

FOIL method

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

Factoring out the GCF

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

Greatest common factor of terms

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

Factoring a trinomial

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

Difference of two squares

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

Perfect square trinomial

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

Rational expression

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

Simplifying a rational expression

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

Excluded value

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

Radical expression

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

Simplifying a square root

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

Product rule for radicals

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

Absolute value

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

Absolute value equation

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

Quadratic equation

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

Standard form of a quadratic equation

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

Zero product property

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

Solving a quadratic by factoring

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

Quadratic formula

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

Discriminant

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

Interpreting the discriminant

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.

Completing the square

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

Root of an equation

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

Equivalent expressions

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

Solving a literal equation

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

Modeling with an equation

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.

Checking a solution

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

Reciprocal

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

Solving a proportion

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

Square root property

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

Monomial

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

Binomial

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

Trinomial

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

Factoring by grouping

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

Solving an inequality

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.

Identity equation

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

Equation with no solution

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

Cross multiplication

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

Function

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

Function notation

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

Evaluating a function

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

Domain of a function

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

Range of a function

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

Vertical line test

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

Independent variable

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

Dependent variable

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

Linear function

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

Slope

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

Positive slope

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

Negative slope

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

Zero slope

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

Undefined slope

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

y-intercept

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

x-intercept

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

Slope-intercept form

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

Point-slope form

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

Standard form of a line

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

Parallel lines

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

Perpendicular lines

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

Quadratic function

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

Parabola

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

Vertex of a parabola

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

Axis of symmetry

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

Vertex form of a quadratic

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

Maximum value of a quadratic

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

Minimum value of a quadratic

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

Zeros of a function

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

Exponential function

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.

Exponential growth

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

Exponential decay

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

Growth factor

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

Average rate of change

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.

Constant rate of change

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

Increasing function

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

Decreasing function

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

Vertical translation

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

Horizontal translation

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

Vertical stretch

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

Reflection across the x-axis

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

Piecewise function

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

Absolute value function

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

Step function

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

Sequence

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

Arithmetic sequence

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

Common difference

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

Arithmetic sequence formula

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

Geometric sequence

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

Common ratio

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

Geometric sequence formula

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

Recursive formula

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

Explicit formula

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

Interpreting a graph

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

Composition of functions

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

Inverse function

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

Linear vs. exponential growth

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

Continuous growth model

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

Inverse variation

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

Direct variation

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

Real number system

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

Rational number

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 $0.25$.

Irrational number

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

Integer

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

Closure under addition

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

Sum of two rationals

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

Sum of a rational and an irrational

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

Product of a nonzero rational and an irrational

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

Commutative property

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

Associative property

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

Order of operations

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

Scientific notation

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

Order of magnitude

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

Rational exponent

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

Dimensional analysis

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

Unit rate

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

Ratio

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

Proportion

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

Percent

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

Reasonableness of an answer

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

Accuracy and precision

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

Mean

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

Median

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

Mode

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

Range of a data set

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

Measure of center

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

Measure of spread

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

Standard deviation

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

Outlier

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

Scatter plot

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

Line of best fit

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

Correlation

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

Positive correlation

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

Negative correlation

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

Correlation versus causation

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

Probability

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

Simple event

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

Complement of an event

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

Compound event

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

Independent events

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

Dependent events

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

Mutually exclusive events

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

Addition rule of probability

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

Multiplication rule for independent events

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

Theoretical probability

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

Experimental probability

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

Sample space

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

Fundamental counting principle

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

Permutation

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

Combination

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

Factorial

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

Histogram

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

Box plot

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

Quartile

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

Interquartile range

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

Frequency

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

Categorical data

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

Numerical data

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

Scale of a graph

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

Origin

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

Significant figures

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

Estimation

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

Absolute value as distance

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

Cube root

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