Mixed algebra review
58 flashcards covering Mixed algebra review for the SAT Math section.
Mixed algebra review involves combining various algebra concepts into a single set of problems, rather than tackling them one at a time. This includes solving equations and inequalities, working with functions, graphing relationships, and manipulating expressions. It's essentially a way to practice how these ideas interconnect in real-world scenarios, helping you build flexibility and problem-solving skills in math.
On the SAT Math section, mixed algebra appears in questions that require you to set up and solve equations from word problems, simplify complex expressions, or analyze functions and their graphs. Common traps include making careless errors in algebraic manipulation, like distributing negatives incorrectly, or overlooking restrictions on variables. Focus on translating words into mathematical terms accurately and double-checking your steps, as these questions often test multiple concepts at once.
A good tip: Always verify your answer by substituting it back into the original problem.
Terms (58)
- 01
Linear equation
A linear equation is an equation that represents a straight line when graphed, typically in the form y = mx + b, where m is the slope and b is the y-intercept.
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Solving linear equations
Solving linear equations involves isolating the variable on one side by performing inverse operations such as addition, subtraction, multiplication, or division to find its value.
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Inequality
An inequality is a mathematical statement that compares two expressions using symbols like <, >, ≤, or ≥, and solving it means finding the values of the variable that make the statement true.
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Compound inequality
A compound inequality combines two inequalities with the word 'and' or 'or', requiring you to solve both parts simultaneously to find the overlapping solution set.
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System of equations
A system of equations is a set of two or more equations with the same variables, and solving it means finding the values that satisfy all equations at once.
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Substitution method
The substitution method for solving systems of equations involves solving one equation for one variable and then substituting that expression into the other equation to solve for the remaining variable.
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Elimination method
The elimination method for solving systems of equations involves adding or subtracting the equations to eliminate one variable, allowing you to solve for the other variable directly.
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Quadratic equation
A quadratic equation is a polynomial equation of degree two, generally in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.
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Factoring quadratics
Factoring quadratics means rewriting a quadratic expression as a product of two binomials, which can help solve equations by setting each factor equal to zero.
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Quadratic formula
The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a), used to find the roots of a quadratic equation ax² + bx + c = 0 when factoring is not straightforward.
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Discriminant
The discriminant is the part of the quadratic formula under the square root, b² - 4ac, which determines the nature of the roots: positive for two real roots, zero for one real root, and negative for no real roots.
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Function
A function is a relation where each input has exactly one output, often denoted as f(x), and it must pass the vertical line test when graphed.
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Domain of a function
The domain of a function is the set of all possible input values for which the function is defined, determined by restrictions like division by zero or square roots of negative numbers.
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Range of a function
The range of a function is the set of all possible output values, found by considering the highest and lowest values the function can produce based on its domain and equation.
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Linear function
A linear function is a function whose graph is a straight line, expressed as f(x) = mx + b, where m is the slope and b is the y-intercept.
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Slope of a line
The slope of a line is a measure of its steepness, calculated as the change in y-values divided by the change in x-values between any two points on the line.
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Y-intercept
The y-intercept is the point where a line crosses the y-axis, represented as the value of b in the equation y = mx + b.
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Parallel lines
Parallel lines are lines in a plane that never intersect and have the same slope, meaning their equations have the same m value but different y-intercepts.
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Perpendicular lines
Perpendicular lines are lines that intersect at a right angle, with slopes that are negative reciprocals of each other.
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Exponential expression
An exponential expression involves a base raised to an exponent, like a^b, and follows rules for multiplication, division, and powers of powers.
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Laws of exponents
The laws of exponents are rules for simplifying expressions with powers, including multiplying powers with the same base by adding exponents and dividing by subtracting exponents.
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Radical expression
A radical expression involves a root, such as a square root or cube root, and can be simplified by factoring out perfect squares or cubes from under the radical.
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Simplifying radicals
Simplifying radicals means reducing them to their simplest form by removing factors that are perfect powers of the root's index, like √(50) simplifies to 5√2.
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Rational exponents
Rational exponents express roots and powers in exponential form, where a^(1/n) means the nth root of a, and a^(m/n) means the nth root of a raised to m.
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Polynomial
A polynomial is an expression with variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents, like 3x^2 + 2x + 1.
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Adding polynomials
Adding polynomials involves combining like terms from each polynomial, such as adding the coefficients of terms with the same degree.
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Multiplying polynomials
Multiplying polynomials requires distributing each term in the first polynomial to every term in the second, then combining like terms.
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Factoring polynomials
Factoring polynomials means rewriting them as a product of simpler expressions, often by finding common factors or using patterns like difference of squares.
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Difference of squares
The difference of squares is a factoring pattern for expressions like a^2 - b^2, which factors into (a - b)(a + b).
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Absolute value equation
An absolute value equation involves an expression inside absolute value bars set equal to a number, and solving it requires considering both positive and negative cases.
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Absolute value inequality
An absolute value inequality compares an absolute value expression to a number using <, >, ≤, or ≥, and solving it involves breaking it into compound inequalities.
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Word problem setup
Word problem setup in algebra means translating real-world scenarios into equations by defining variables for unknowns and using relationships to form equations.
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Rate, time, distance formula
The rate, time, distance formula is distance = rate × time, used to solve problems involving motion, such as finding speed or time traveled.
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Work rate problems
Work rate problems involve rates of work, like jobs per hour, and solving them often requires adding rates when multiple entities work together.
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Mixture problems
Mixture problems involve combining solutions with different concentrations to achieve a desired mixture, solved by setting up equations based on total amounts and concentrations.
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Percent change
Percent change calculates the relative increase or decrease between two values, using the formula [(new value - original value) / original value] × 100%.
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Arithmetic sequence
An arithmetic sequence is a sequence of numbers where the difference between consecutive terms is constant, and the nth term is found using an = a1 + (n-1)d.
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Geometric sequence
A geometric sequence is a sequence where each term is multiplied by a constant ratio to get the next, with the nth term given by an = a1 × r^(n-1).
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Extraneous solutions
Extraneous solutions are values that satisfy an equation derived from the original but not the original itself, often occurring in equations with square roots or absolute values.
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Graphing linear equations
Graphing linear equations involves plotting points that satisfy the equation and drawing a straight line through them, using intercepts or slope for accuracy.
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X-intercept
The x-intercept is the point where a line crosses the x-axis, found by setting y = 0 in the equation and solving for x.
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Function composition
Function composition is combining two functions by using the output of one as the input of the other, denoted as f(g(x)), and evaluated step by step.
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Inverse function
An inverse function reverses the operation of the original function, so if f(x) = y, then f⁻¹(y) = x, and it must pass the horizontal line test.
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Rational equation
A rational equation is an equation containing fractions with variables in the denominator, solved by multiplying both sides by the common denominator to eliminate fractions.
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Completing the square
Completing the square is a method to rewrite a quadratic equation in the form (x + p)^2 = q by adding and subtracting a constant to make a perfect square trinomial.
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Vertex form of quadratic
The vertex form of a quadratic is y = a(x - h)^2 + k, where (h, k) is the vertex, useful for identifying the maximum or minimum point of a parabola.
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Axis of symmetry
The axis of symmetry of a parabola is a vertical line that passes through its vertex, given by x = -b/(2a) for a quadratic y = ax^2 + bx + c.
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Order of operations
The order of operations is a rule that dictates the sequence for evaluating expressions: parentheses first, then exponents, multiplication and division from left to right, and addition and subtraction from left to right.
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Distributive property
The distributive property states that a(b + c) = ab + ac, allowing you to multiply a term through parentheses to simplify expressions.
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Combining like terms
Combining like terms means adding or subtracting the coefficients of terms that have the same variables raised to the same powers, such as 3x + 2x = 5x.
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Fraction operations in algebra
Fraction operations in algebra involve adding, subtracting, multiplying, or dividing fractions with variables, requiring a common denominator for addition and subtraction.
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Proportion solving
Proportion solving means finding an unknown in a ratio set equal to another ratio, often by cross-multiplying and solving the resulting equation.
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Similar figures algebra
Similar figures algebra involves setting up proportions based on corresponding sides of similar shapes, like triangles, to solve for unknown lengths.
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Break-even point
The break-even point is the level of production or sales where total costs equal total revenue, found by solving an equation where cost equals income.
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Asymptotes of rational functions
Asymptotes of rational functions are lines that the graph approaches but never touches, including vertical asymptotes at values making the denominator zero and horizontal ones based on degrees.
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Solving rational inequalities
Solving rational inequalities involves finding critical points where the expression equals zero or is undefined, then testing intervals to determine where the inequality holds.
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Logarithms
Logarithms are the inverse of exponential functions, where logb(x) = y means b^y = x, and they follow properties like the product rule for adding logs.
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Exponential growth
Exponential growth is a process where a quantity increases by a fixed percentage over equal intervals, modeled by equations like y = a(1 + r)^t.