91Ó°ÊÓ

Linear inequalities are expressions that involve linear terms and compare two values with an inequality sign. In a linear inequality like the one in our example, which is written as \(2x - 3v \geq 6\), the goal is to find the range of values that \(x\) can take to satisfy the inequality. Unlike linear equations, which have only one solution, linear inequalities describe a range of solutions.

To solve linear inequalities, we use similar techniques to solving linear equations, such as adding or subtracting the same value to both sides, and multiplying or dividing both sides by the same non-zero number. However, there's one key difference: when we multiply or divide both sides by a negative number, the inequality sign must be flipped to maintain the true statement.

Inequality manipulation involves operations that maintain the inequality's truth, while aiming to isolate the variable of interest on one side. These operations can include addition, subtraction, multiplication, and division. However, it's important to pay attention to the rules unique to inequalities—especially the rule that if you multiply or divide both sides by a negative number, the direction of the inequality sign reverses.

Let's go through our textbook example: starting with \(2x - 3v \geq 6\), we bring the term \(3v\) to the other side by adding \(3v\) to both sides. Then, to solve for \(x\), we divide both sides by 2, which is a positive number, so the inequality sign remains the same. The careful manipulation of the inequality leads us to the correct solution, which illustrates the set of values for \(x\) in relation to \(v\).

Algebraic expressions are combinations of numbers, variables, and mathematical operators. They do not have inequality or equality signs, hence they do not form equations or inequalities. However, when we work with inequalities, such as \(2x - 3v \geq 6\), we manipulate algebraic expressions within those inequalities to solve for the variable.

In the given problem, we treat the inequality almost as we would an algebraic equation, performing operations to both sides to isolate the variable of interest. Our algebraic expression after the first step is \(2x \geq 6 + 3v\), which uses addition to combine constants and variables. The final rearranged expression represents our solution, \(x \geq 3 + \frac{3}{2}v\), now in a format that's easy to understand and use for graphing or further calculations.