Integers, Fractions, and Absolute Value

Objective: Use sign rules and number-line reasoning.

An integer may be positive, negative, or zero. Adding a negative number moves left on a number line; subtracting a number is equivalent to adding its additive inverse. Thus −4−(−7)=−4+7=3. Products and quotients of two nonzero numbers with the same sign are positive; opposite signs give negative. Zero multiplied by any number is zero, whereas division by zero is undefined. Sign rules should be paired with magnitude checks, especially for sums with opposite signs.

Objective: Operate on fractions using common denominators and reciprocals.

A rational number can be written a/b where a and b are integers and b≠0. Fractions must use a common denominator for addition or subtraction: 1/3+1/6=2/6+1/6=1/2. Multiply numerators and denominators for fraction products; divide by a nonzero fraction by multiplying by its reciprocal. A decimal that terminates or repeats is rational. Fractions can be negative, and negative signs may be moved consistently between numerator, denominator, and outside the fraction. Simplify results when possible and estimate to detect implausible answers.

Objective: Interpret absolute value and distinguish opposite numbers.

Absolute value abs(x) is the distance from x to zero on the real number line and is nonnegative. Thus abs(−6)=6, |0|=0, and −abs(−6)=−6; the outside negative sign is not part of the absolute value. The distance between a and b is abs(a−b), which equals abs(b−a). Absolute value is not a rule that simply erases a negative sign in every surrounding expression. In introductory contexts, abs(x)=4 has two solutions, x=4 or x=−4, while abs(x)=−4 has no real solution.