Quick Algebra Review

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Questions and Answers

What is an expression in mathematics?

A mathematical "phrase" that contains numbers and variables but not an equal sign.

What is the first thing you should do when simplifying an expression?

Combine like terms.

What does the acronym PEMDAS stand for?

Please Excuse My Dear Aunt Sally

What is the order of operations in PEMDAS?

<p>Parenthesis, Exponents, Multiplication/Division (from left to right), Addition/Subtraction (from left to right)</p> Signup and view all the answers

What is the goal of solving equations?

<p>To get the variable by itself (solve for x).</p> Signup and view all the answers

What is the important rule to remember when solving equations?

<p>Always do to one side of the equal sign what you do to the other.</p> Signup and view all the answers

What are the three steps involved in translating a word problem into an equation?

<p>Understand the problem, define the variables, write the equation</p> Signup and view all the answers

What does it mean to solve an inequality?

<p>To find all the values of the variable that make the inequality true.</p> Signup and view all the answers

What is the rule to remember when multiplying or dividing both sides of an inequality by a negative number?

<p>Flip the sign of the inequality.</p> Signup and view all the answers

What is the absolute value of a number?

<p>The distance of the number from zero on the number line.</p> Signup and view all the answers

What is the definition of a linear equation?

<p>An equation whose graph is a straight line.</p> Signup and view all the answers

What are the two important things to know when writing the equation of a line in slope-intercept form?

<p>The slope and the y-intercept.</p> Signup and view all the answers

What is the slope of a line?

<p>The rate of change, represented as rise over run.</p> Signup and view all the answers

What is the y-intercept of a line?

<p>The point where the line crosses the y-axis.</p> Signup and view all the answers

What is a system of equations?

<p>A set of two or more equations that must be solved simultaneously to find a solution that satisfies all of them.</p> Signup and view all the answers

What is the solution to a system of equations when using the graphing method?

<p>The point where the graphs of the equations intersect.</p> Signup and view all the answers

Explain the substitution method for solving systems of equations.

<p>Solving one equation for one variable and then substituting that expression into the other equation.</p> Signup and view all the answers

What are the laws of exponents?

<p>Rules that govern how exponents work when performing operations like multiplication, division, and raising powers to powers.</p> Signup and view all the answers

What does it mean to simplify an expression involving exponents?

<p>To combine terms with the same base and exponent.</p> Signup and view all the answers

What happens when multiplying exponents with the same base?

<p>Add the exponents together.</p> Signup and view all the answers

What happens when raising a power to another power?

<p>Multiply the exponents.</p> Signup and view all the answers

What happens when dividing exponents with the same base?

<p>Subtract the exponents.</p> Signup and view all the answers

What is the value of any number raised to the power of zero?

<ol> <li></li> </ol> Signup and view all the answers

What does a negative exponent indicate?

<p>The inverse of the base.</p> Signup and view all the answers

What is the definition of a quadratic equation?

<p>An equation that contains a variable raised to the second power (x²).</p> Signup and view all the answers

What is the goal when factoring a quadratic equation?

<p>To find two terms that multiply together to equal zero.</p> Signup and view all the answers

What is the quadratic formula?

<p>A formula used to solve quadratic equations that are not easily factorable.</p> Signup and view all the answers

What is a rational expression?

<p>An expression that can be written as a fraction where the variable appears in the denominator.</p> Signup and view all the answers

What is the domain of a rational expression?

<p>The set of all real numbers except for those that make the denominator equal to zero.</p> Signup and view all the answers

What is the first step in simplifying a rational expression?

<p>Factor the numerator and denominator, if possible.</p> Signup and view all the answers

How do you multiply rational expressions?

<p>Multiply the numerators and denominators straight across, and cancel any common factors.</p> Signup and view all the answers

How do you divide rational expressions?

<p>Multiply the first expression by the reciprocal of the second expression.</p> Signup and view all the answers

What is the rule to remember when adding or subtracting rational expressions?

<p>They must have a common denominator.</p> Signup and view all the answers

How do you solve a rational equation?

<p>Multiply both sides of the equation by the least common denominator, and then solve as a regular equation, making sure to check for extraneous solutions.</p> Signup and view all the answers

What is the rule regarding even roots of negative numbers?

<p>Even roots of negative numbers are undefined, as there are no real numbers that multiply by themselves an even number of times to produce a negative result.</p> Signup and view all the answers

What are fractional exponents?

<p>Exponents that are written as fractions.</p> Signup and view all the answers

How are fractional exponents related to radicals?

<p>The denominator of a fractional exponent indicates the root, and the numerator indicates the power to be applied to the radical.</p> Signup and view all the answers

What is the first step in solving a radical equation?

<p>Isolate the radical on one side of the equation.</p> Signup and view all the answers

Flashcards

Expression

A mathematical 'phrase' that contains numbers and variables, but not an equal sign.

Equation

A mathematical statement that two expressions are equal, containing an equal sign (=).

Simplifying Expressions

The process of simplifying an expression to have as few terms as possible, making it easier to work with.

Order of Operations (PEMDAS)

A set of rules used to determine the order in which operations are performed in an expression or equation.

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Solving Equations

The process of finding the value of a variable that makes an equation true.

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Word Problems

Problems presented in written language that require translation into mathematical equations.

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Inequality

A mathematical statement that compares two expressions using inequality symbols (<, >, ≤, ≥).

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Absolute Value

The distance of a number from zero on the number line, always positive.

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Linear Equation

An equation whose graph is a straight line.

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Slope

The rate of change of a linear function, calculated as rise over run.

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Y-intercept

The point where a line crosses the y-axis.

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System of Equations

A set of two or more equations that share the same variables and have a common solution.

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Solving by Graphing

Finding the solution to a system of equations by graphing the equations and identifying the point of intersection.

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Solving by Substitution

Finding the solution to a system of equations by solving one equation for a variable and substituting it into the other equation.

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Laws of Exponents

Rules for simplifying expressions involving exponents.

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Polynomial Multiplication

A product of two binomials, often expanded using the FOIL method.

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Quadratic Equation

An equation that includes a variable raised to the second power (x²).

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Solving by Factoring

Finding solutions to a quadratic equation by factoring the expression into two binomials.

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Quadratic Formula

A formula that provides the solutions to any quadratic equation of the form ax² + bx + c = 0.

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Rational Expression

An expression that can be written as a fraction with variables in the denominator.

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Domain of Rational Expressions

The set of all possible values of a variable that make a rational expression defined (avoiding division by zero).

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Simplifying Rational Expressions

Simplifying a rational expression by factoring both numerator and denominator, canceling like factors, and removing any values not in the domain.

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Multiplying Rational Expressions

Multiplying rational expressions by multiplying the numerators and denominators, then simplifying.

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Dividing Rational Expressions

Dividing rational expressions by multiplying the first expression by the reciprocal of the second expression.

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Adding/Subtracting Rational Expressions

Adding or subtracting rational expressions by first finding a common denominator.

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Solving Rational Equations

Solving an equation containing rational expressions by multiplying both sides by the least common denominator.

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Radical Expression

An expression involving roots (like square roots, cube roots, etc.) of variables.

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Rule for Even Roots

A rule for radical expressions stating that even roots cannot have negative numbers under them.

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Fractional Exponents

A way to represent radicals using fractional exponents, where the denominator represents the root and the numerator represents the power.

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Solving Radical Equations

Solving an equation containing a radical by raising both sides of the equation to the power of the radical, then solving for the variable.

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Study Notes

A Quick Algebra Review

  • Simplifying Expressions: An expression is a mathematical phrase, not an equation. It contains numbers and variables, but no equal sign. Equations contain an equal sign. Combining like terms is the first step in simplifying an expression.

  • Solving Equations: The goal is to isolate the variable. Reverse the order of operations. Perform addition and subtraction first, then multiplication and division, and finally exponents. Ensure that operations performed on one side of the equation are also performed on the other.

  • Problem Solving:

  • Understand the problem

  • Define the variables

  • Write an equation

  • Inequalities: Use proper inequality symbols to represent relationships between values. Reversing the inequality sign occurs during multiplication or division by a negative number.

  • Absolute Values: The absolute value of a number is its distance from zero on the number line. Absolute values are always positive.

  • Linear Equations: Linear equations graph as a straight line. Slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept.

  • Systems of Equations: A system of equations is two or more equations considered together. Graphing or substitution can be used to solve systems. The solution is the point of intersection of the lines.

  • Laws of Exponents: Rules for dealing with powers of variables and numbers with the same base.

  • Multiplying: add exponents

  • Raising a power to a power: multiply exponents

  • Dividing: subtract exponents

  • Zero exponent: any number to the zero power is 1

  • Negative exponent: reciprocal of the base to the positive power

  • Quadratics: Equations containing a variable to the second power (x²). Factoring and the quadratic formula are methods for solving.

  • Rational Expressions: Expressions written as fractions with variables.

  • The denominator can never be zero.

  • Factor numerator and denominator

  • Find any values that make the denominator zero. Rewrite to show a fraction of 1 , then cancel like factors

  • Radical Expressions: Expressions with roots of variables, including square roots and cube roots.

  • A square root can't have a negative number inside.

  • Fractional Exponents: Exponents in the form of fractions, where the denominator indicates what root to take, and the numerator indicates the power to raise the root to.

  • Solving Radical Equations:

  • Isolate the radical.

  • Raise both sides to the power of the radical.

  • Check solution in the original equation to verify it is part of the domain, as some solutions may not make sense for a particular expression.

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