INTEGRERS

Absolute Value of Integers

The absolute value of an integer refers to its distance from zero on the number line, which is always expressed as a positive number. For instance, the absolute value of $-2$ is written as $|-2| = 2$, while the absolute value of 6 is $|6| = 6$. This rule applies to all numbers: the absolute value of $-5$ is 5, the absolute value of 0 is 0, and the absolute value of 3 is 3.

Adding Integers

When adding integers with the same sign, you add the absolute values and keep the common sign. For example, $2 + 5 = 7$, while $-3 + (-2) = -5$ and $-4 + (-1) = -5$. If the integers have different signs, you subtract the smaller absolute value from the larger one and apply the sign of the integer with the larger absolute value. This is demonstrated in examples such as $5 + (-2) = 3$, $2 + (-6) = -4$, and $2 + (-4) = -2$.

Subtracting Integers

To subtract integers when they have the same signs, change the sign of the minuend and subtract, then copy the sign of the larger absolute value. For instance, in the subtraction $2 - 5$, if you change 2 to $-2$, the problem becomes $(-2) - 5 = -3$. When the signs are different, you add the absolute values and use the sign of the larger absolute value. Examples of this include $9 - 4 = 5$ or $(-4) - (-3) = -1$, where the matching signs lead to the subtraction of absolute values.

Multiplying & Dividing Integers

The rules for multiplication and division depend on whether the signs of the integers match. If both integers have the same sign, the result is always positive, such as in $(-1) \times (-5) = 5$ or $(+2) \times (+3) = 6$. Conversely, if the signs are different, the result will always be negative, as seen in examples like $(-5) \times (+3) = -15$ or $(+6) \times (-2) = -12$.

Order of Operations with Integers (GEMDAS)

The GEMDAS acronym provides the order of operations for solving integer expressions: Grouping (parentheses, brackets, or braces), Exponents, Multiplication, Division, Addition, and Subtraction. Simple calculations follow this flow, such as $6 \times 3 \times 4 = 72$. When applying GEMDAS to a more complex expression like $(2 + 4) \times 3 - 8 \times 2$, you first solve the grouping to get $6 \times 3 - 8 \times 2$, then perform multiplications to get $18 - 16$, resulting in a final answer of 2.

Algebraic Expressions

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and operations, which distinguishes it from arithmetic. Within algebra, constants are fixed values like $7, -2$, or $22$ that do not change, while variables such as $x, y$, or $z$ serve as symbols for different values. Algebraic expressions, such as $x + 3$ or $2x + y$, combine variables, constants, and operations without using an equal sign.

Variables, Constants, and Terms

An algebraic expression is composed of terms, which are separate elements divided by addition or subtraction. For example, in the expression $3x + 5$, both $3x$ and $5$ are individual terms. Within these terms, a coefficient is the number that multiplies a variable; for instance, the numeric coefficient of $x$ in $3x$ is 3. In an expression like $7x - 4$, $x$ is the variable, $-4$ is the constant, there are two terms, and the operation being performed is subtraction.

Algebraic Phrases and Their Expressions

Common verbal phrases can be translated into algebraic expressions using key mathematical operations. Addition is indicated by terms such as "the sum of," "increased by," or "more than," allowing "the sum of x and 9" to be written as $x + 9$. Subtraction uses phrases like "the difference between" or "decreased by" to form expressions like $x - 5$. Multiplication appears as "the product of" or "twice a number," translating to $6x$ or $2x$, while division is marked by "the quotient of" or "half of a number," resulting in $x \div 4$ or $\frac{1}{2}x$.

Evaluating Algebraic Expressions

Evaluating an expression involves substituting given values for variables and then simplifying the result following the GEMDAS order of operations. For example, to evaluate $2x + 3$ when $x = 4$, you substitute to get $2(4) + 3$, resulting in $8 + 3 = 11$. Similarly, with $x = 4$ and $y = 6$, the expression $x^2 + y$ becomes $4^2 + 6$, which simplifies to $16 + 6 = 22$.

Real-world Problems Involving Algebraic Expressions

Algebraic expressions help solve real-world problems like calculating profit, distance, cost, and temperature. A business can find profit using $20x - 200$, yielding Php 100 if 15 units are sold. Distance traveled at a specific speed over time can be calculated with $50t + 100$, resulting in 250 miles after 3 hours. Cost-related formulas like $10x + 300$ for pizza toppings and temperature calculations like $20(2) + 100$ for an oven also rely on these algebraic principles to find specific results.

Polynomial Expressions

A polynomial is an algebraic expression with one or more terms, each containing constants, variables, or coefficients. These are classified by the number of terms: monomials have one term, binomials have two, trinomials have three, and those with four are quadrinomials, while expressions with five or more are multinomials. Polynomials are also described by their degree, which is the sum of the exponents in a term. This includes constant polynomials with a degree of zero, linear polynomials with a degree of one, quadratic at degree two, cubic at degree three, quartic at four, and quintic at degree five.