math

solve

(1) To find all the solutions to an equation or an inequality (or a system of equations or inequalities). For example, solving the equation x2 = 9 gives the solutions x = 3 and x = −3.

equation

A mathematical sentence in which two expressions appear on either side of an "equals" sign (=), stating that the two expressions are equivalent. For example, the equation 7x + 4.2 = −8 states that the expression 7x + 4.2 has the value -8. In this course, an equation is often used to represent a rule relating two quantities. For example, a rule for finding the area y of a tile pattern with figure number x might be written y = 4x − 3.

negative

A negative number is a number less than zero. Negative numbers are graphed on the negative side of a number line.

zero

A number often used to represent "having none of a quantity." Zero is neither negative nor positive. Zero is the additive identity.

variable

A symbol used to represent one or more numbers. In this course, letters of the English alphabet are used as variables. For example, in the expression 3x − (8.6xy + z), the variables are x, y, and z.

term

A term is a single number, variable, or the product of numbers and variables. In an expression, terms are separated by addition or subtraction signs. For example, in the expression 1.2x - 45 + 3xy2, the terms are 1.2x, -45, and 3xy2. A term is also a component of a sequence.

algebra tiles

An algebra tile is a manipulative whose area represents a constant or variable quantity. The algebra tiles used in this course consist of large squares with dimensions x-by-x and y-by-y; rectangles with dimensions x-by-1, y-by-1, and x-by-y; and small squares with dimensions 1-by-1. These tiles are named by their areas: x2, y2, x, y, xy, and 1, respectively. The smallest squares are called "unit tiles." In this text, shaded tiles will represent positive quantities while unshaded tiles will represent negative quantities.

expression

An expression is a combination of individual terms separated by plus or minus signs. Numerical expressions combine numbers and operation symbols; algebraic (variable) expressions include variables. For example, 4 + (5 − 3) is a numerical expression. In an algebraic expression, if each of the following terms, 6xy2, 24, and are combined, the result may be 6xy2 + 24 − . An expression does not have an "equals" sign.

order of operations

An expression is a combination of individual terms separated by plus or minus signs. Numerical expressions combine numbers and operation symbols; algebraic (variable) expressions include variables. For example, 4 + (5 − 3) is a numerical expression. In an algebraic expression, if each of the following terms, 6xy2, 24, and are combined, the result may be 6xy2 + 24 − . An expression does not have an "equals" sign.

Expression Mat

An organizing tool used to visually represent an expression with algebra tiles.

equation mat

An organizing tool used to visually represent two equal expressions using algebra tiles. For example, the equation mat below represents the equation 2x − 1 − (−x + 3) = 6 − 2x.

combining like terms

Combining two or more like terms simplifies an expression by summing constants and summing those variable terms in which the same variables are raised to the same power. For example, combining like terms in the expression 3x + 7 + 5x − 3 + 2x2 + 3y2 gives 8x + 4 + 2x2 + 3y2. When working with algebra tiles, combining like terms involves putting together tiles with the same dimensions.

area

For this course, area is the number of square units needed to fill up a region on a flat surface. In later courses, the idea will be extended to cones, spheres, and more complex surfaces.

solution

The number or numbers that when substituted into an equation or inequality make the equation or inequality true. For example, x = 4 is a solution to the equation 3x − 2 = 10 because 3x − 2 equals 10 when x = 4. A solution to a two-variable equation is sometimes written as an ordered pair (x, y). For example, x = 3 and y = −2 is a solution to the equation y = x − 5; this solution can be written as (3, −2).

evaluate

To evaluate an expression, substitute the value(s) given for the variable(s) and perform the operations according to the order of operations. For example, evaluating 2x + y − 10 when x = 4 and y = 3 gives the value 1.

simplify

To simplify an expression is to write a less complicated expression with the same value. A simplified expression has no parentheses and no like terms. For example, the expression 3 − (2x + 7) − 4x can be simplified to −4 − 6x. When working with algebra tiles, a simplified expression uses the fewestpossible tiles to represent the original expression.

opposite

Two numbers are opposites if they are the same distance from zero, but one is positive and one is negative. For example, 5 and −5 are opposites. The opposite of a number is sometimes called its additive inverse, indicating that the sum of a number and its opposite is zero.

equal

Two quantities are equal when they have the same value. For example, when x = 4, the expression x + 8 is equal to the expression 3x because their values are the same.

greater

of an extent, amount, or intensity considerably above the normal or average.

minus

with the subtraction of.