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System of linear inequalities

A set of two or more inequalities.

 

System of linear equations

A set of two or more equations.

 

Sum of difference of two cubes

Let $a$ and $b$ be real numbers, variables, or algebraic expressions.
1. \({\rm{ }}{a^3} + {b^3} = \left( {a + b} \right)\left( {{a^2} – ab + {b^2}} \right)\) 2. \({\rm{ }}{a^3} – {b^3} = \left( {a – b} \right)\left( {{a^2} + ab + {b^2}} \right)\)

 

Sum and difference of two terms

Let $a$ and $b$ be real numbers, variables, or algebraic expressions.
\(\left( {a + b} \right)\left( {a – b} \right) = {a^2} – {b^2}\)

 

Sum

The result when two or more numbers are added.

 

Subtracting one integer from another

Add the opposite of the integer being subtracted to the other integer.

 

Subtracting fractions with unlike denominators

Rewrite the fractions so that they have like denominators. Then use the rule for subtracting fractions with like denominators.

 

Subtracting fractions with like denominators

Let $a$, $b$, and $c$ be integers with $c$\( \ne \)0. Then use the following rule: \(\frac{a}{c} – \frac{b}{c} = \frac{{a – b}}{c}\).

 

Subtraction property of inequalities

Subtract the same quantity form each side. If \(a < b\) , then \(a + c < b + c\).

 

Standard form of a polynomial

Let \({a_n},{\rm{ }}{a_{n – 1}},{\rm{ }}…,{\rm{ }}{a_2},{\rm{ }}{a_1},{\rm{ }}{a_0}\) be real numbers and let n be a non negative number. Order the terms with descending exponents.

\[{a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + … + {a_2}{x^2} + {a_1}x + {a_0}\]

 

Standard form of an equation in one variable

Let \({a_n},{\rm{ }}{a_{n – 1}},{\rm{ }}…,{\rm{ }}{a_2},{\rm{ }}{a_1},{\rm{ }}{a_0}\) be real numbers and let n be a nonnegative number. Order the terms with descending exponents.

\[{a_n}{x^n} + {a_{n – 1}}{x^{n – 1}} + … + {a_2}{x^2} + {a_1}x + {a_0} = 0\]

 

Standard form of a complex number

Let $a$ and $b$ be real numbers. \(a + bi\)

 

Squaring property of equality

Let $a$ and $b$ be real numbers, variables, or algebraic expressions. If $a = b$, then it follows that $a^2 = b^2$.

 

Square root property (complex square roots)

The equation \({u^2} = d,{\rm{ }}d < 0\), has exactly two solutions: \(u = \sqrt {\left| d \right|} i\) and \(u = - \sqrt {\left| d \right|} i\). These solutions can also be written as \(u = \pm \sqrt {\left| d \right|} i\).

 

Square root property

Let $u$ be a real number, a variable, or an algebraic expression, and let $d$ be a positive real number; then the equation $u^2 = d$ has exactly two solutions.
If $u^2 = d$, then \(u = \sqrt d \) and \(u = – \sqrt d \). These solutions can also be written as \(u = \pm \sqrt d \).

 

Square root of x squared

If $x$ is a real number, then \(\sqrt {{x^2}} = \left| x \right|\). For a special case in which you know that $x$ is a non-negative real number, you can write \(\sqrt {{x^2}} = x\).

 

Square root

When $n = 2$ in \(a = {b^n}\).

 

Square of a binomial

Let $a$ and $b$ be real numbers, variables, or algebraic expressions. \({\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\) \({\left( {a – b} \right)^2} = {a^2} – 2ab + {b^2}\)

 

Solve an inequality

Finding all values of the variable(s) for which an inequality set is true.

 

Solve

Find all values of the variable(s) for which the equation is true.

 

Solutions

The values of the variable(s) for which an equation is true.

 

Solution set

The set of all real numbers that are solutions of an inequality.

 

Slope-intercept form

The form of the equation \(y = mx + b\) where the slope of the line is $m$ and the $y$-intercept is \(\left( {0,b} \right)\).

 

Slope

The change in $y$ divided by the change in $x$ of two points.

 

Simplifying rational expressions

Let $a$, $b$, and $c$ represent real numbers, variables, or algebraic expressions such that \(b \ne 0\) and \(c \ne 0\). Then the following is valid. \(\frac{{ac}}{{bc}} = \frac{a}{b}\).

 

Simplifying radical expressions

A radical expression is said to be in simplest form when all three of the following statements are true. 1. All possible $n$th-powered factors have been removed from each radical. 2. No radical contains a fraction. 3. No denominator of a fraction contains a radical.

 

Simplify an algebraic expression

Remove symbols of grouping and combine like terms.

 

Second-degree polynomial equations

 

Scientific notation

Writing a number, usually very large or very small, in the form \(c \times {10^n}\), where \(1 \le c < 10\) and $n$ is an integer.

 

Satisfy

The solutions of an equation are said to satisfy the equation when the equation yields a true statement.

 
Glossary: S

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