Linear Expressions, Equations & Inequalities Portfolio.
In this portfolio, we will explore linear expressions, equations, and inequalities. We will discuss how to identify and simplify linear expressions, solve linear equations, and solve linear inequalities. Additionally, we will examine real-world applications of linear expressions, equations, and inequalities.
1. Linear Expressions:
A linear expression is an algebraic expression that is comprised of variables, constants, and coefficients that are only multiplied by a variable (not squared or cubed). For example, the expression 3x + 2y is a linear expression.
To simplify a linear expression, we combine like terms by adding or subtracting coefficients of the same variable. For example, to simplify the expression 4x + 2x - 7, we combine the coefficients of x to get 6x - 7.
2. Linear Equations:
A linear equation is an algebraic equation that represents a straight line on a graph. It can be written in the form ax + by = c, where a, b, and c are constants and x and y are variables. To solve a linear equation, we use the properties of equality to isolate the variable.
For example, to solve the equation 2x + 5 = 11, we subtract 5 from both sides to get 2x = 6, and then divide by 2 to get x = 3.
3. Linear Inequalities:
A linear inequality is an algebraic inequality that represents a range of possible values for a variable. It can be written in the form ax + b < c, ax + b > c, ax + b ≤ c, or ax + b ≥ c, where a, b, and c are constants and x is a variable. To solve a linear inequality, we use similar methods as solving linear equations, but we must also consider the direction of the inequality symbol.
For example, to solve the inequality 3x - 4 ≤ 8, we add 4 to both sides to get 3x ≤ 12, and then divide by 3 to get x ≤ 4.
4. Real-World Applications:
Linear expressions, equations, and inequalities are commonly used in real-world scenarios, such as calculating sales revenue, determining profit margins, and analyzing production costs. By understanding and applying the principles of linear algebra, we can make informed decisions in various business and financial contexts.
In this portfolio, we have discussed the basics of linear expressions, equations, and inequalities, as well as their real-world applications. By mastering these concepts, we can enhance our problem-solving skills and effectively navigate mathematical challenges in different contexts.