91Ó°ÊÓ

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In this exercise, we're using algebra to solve inequalities. Inequalities use symbols like ">", "<", ">=", and "<=" to compare values. These are symbolic representations that show how numbers or expressions relate to each other. Algebraic techniques allow us to simplify and solve problems involving these relationships. By performing operations such as addition, subtraction, multiplication, and division on both sides of an inequality, we can solve for the unknown variable. Understanding algebra helps in recognizing patterns and constructing equations or inequalities that represent real-world situations. When solving inequalities like the one given, we maintain the balance of the expression, much like in solving equations.

Inequality manipulation involves performing operations on inequalities to isolate the variable. This is similar to solving equations, but with an added rule: if you multiply or divide both sides by a negative number, you must reverse the inequality sign. Let's break it down:

• In the first step, we moved all terms involving \(x\) to one side. By adding \(x\) to both sides, we canceled out the negative \(x\) on the left.
• Next, we dealt with the constants. By adding 11 to both sides, we moved the constant on the right to simplify the expression.

Each step maintains the inequality's truth, meaning the relationship between the sides stays the same. This process helps show exactly what value \(x\) can take to satisfy the inequality. These techniques highlight how inequality manipulation is crucial for finding solutions step-by-step, ensuring logical and balanced changes throughout.

Linear inequalities like \(-x-5<3x-11\) are expressions with a straight-line graph representing solutions. They contain variables raised to the first power, making them easier to handle compared to quadratic or higher-degree inequalities.

• The given inequality is a linear one, with the variable \(x\) having the power of one.
• In solving it, operations were used to isolate \(x\), showing all possible values it can take.

The solution \(x > \frac{3}{2}\) tells us that \(x\) must be greater than \(\frac{3}{2}\) to satisfy the condition. Linear inequalities have solutions that can be plotted on a number line, showing which regions satisfy the inequality. This graphical representation and logical steps make them a fundamental part of algebra, helping solve real-world problems involving constraints and conditions.