Solving Linear Equations and Inequalities

Vocabulary flashcards covering like terms, multi-step linear equations, fraction and decimal operations, and rules for solving and graphing inequalities based on class lecture notes.

Like Terms

Terms in an algebraic expression that have identical variable parts raised to the same power, such as $2x$ and $10x$, or $2x^2$ and $6x^2$, which can be combined by adding or subtracting their coefficients.

Combining Like Terms

The process of simplifying an algebraic expression or equation by grouping and performing addition or subtraction on terms with identical variables and exponents, such as combining $8x - 4x$ to get $4x$, or $-5x - 8x$ to get $-13x$.

Multi-Step Equations

Algebraic equations requiring two or more inverse operations to solve, frequently involving applying the distributive property, combining like terms, and isolating the variable (e.g., $3(x + 2) - 8 = 10$).

Distributive Property in Equations

The algebraic property of multiplying a factor outside of parentheses across each term inside parentheses before combining terms, illustrated in steps such as expanding $3(x + 2)$ to $3x + 6$ or $4(x - 2)$ to $4x - 8$.

Inverse Operations

Opposite mathematical operations used to isolate a variable on one side of an equation, such as using subtraction to cancel addition (e.g., subtracting $8$ from both sides of $x + 8 = 16$ to find $x = 8$), or division to cancel multiplication.

Variables on Both Sides

An equation format where variable terms appear on both the left and right sides of the equal sign, requiring inverse operations to collect all variable terms onto a single side (such as subtracting $2x$ from both sides of $4x = 2x + 4$ to get $2x = 4$).

Solving Equations with Fractions

The method of isolating a variable with a fractional coefficient by multiplying both sides of the equation by the denominator or reciprocal, as seen in $\frac{2}{3}x = 10$, where multiplying yields $x = 15$.

Adding Mixed Numbers with Unlike Denominators

The process of converting mixed numbers into improper fractions, converting to a common denominator, adding the numerators, and simplifying back to mixed form, as shown in $1\frac{1}{3} + 2\frac{1}{6} = \frac{8}{6} + \frac{13}{6} = \frac{21}{6} = 3\frac{1}{2}$.

Decimal and Percent Equations

Linear equations involving percent rates or decimal coefficients, such as $P + 0.07P = 60$, which simplifies to $1.07P = 60$ and solves to approximately $P = 56.07$ by dividing both sides by $1.07$.

Inequality Negative Coefficient Rule

The mathematical rule specifying that whenever both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality sign must be reversed (switched).

Graphing an Inequality on a Number Line

A visual representation showing all values that satisfy an inequality, such as using an open circle at $6$ with an arrow drawn to the left to denote $x < 6$ (or $6 > x$).

Number of Solutions: Equations vs. Inequalities

The principle that standard linear equations yield exactly $1$ solution, whereas inequalities yield more than $1$ or many solutions representing an entire set or range of values.