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Interval notation is a useful and concise way of writing sets of numbers—typically solutions to inequalities. Understanding it allows you to easily describe a range of numbers that satisfy a particular condition. In interval notation, you use brackets and parentheses to denote closed and open ends of an interval, respectively. A closed end, indicated by a square bracket [], means the number at that end is included in the set, while an open end, marked by a parenthesis (), implies the number is not included.

For example, the solution to the given inequality is expressed as \(\left(\frac{5}{9},\infty\right)\), where the parenthesis around \(\frac{5}{9}\) means this number is not included in the set, and the interval extends to infinity, which is always represented with a parenthesis since infinity cannot be reached. Thus, any number greater than \(\frac{5}{9}\) is considered a solution.

Linear inequalities are similar to linear equations but instead of an equal sign, they use inequality symbols: \(<\), \(>\), \(\leq\), or \(\geq\). Solving a linear inequality means finding all values of the variable that make the inequality true. The procedure resembles solving a linear equation: you perform algebraic operations to isolate the variable, while remembering some key differences, such as the rule that when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Working with our textbook problem, the initial inequality is \(5 - 7x < 2x - 4\). After using algebraic manipulations, we find that all numbers greater than \(\frac{5}{9}\) make the inequality true. These solutions are then represented on a number line where direction matters - to the right for greater than, to the left for less than. This visual can be converted into interval notation, as seen in the solution section.

Mastering algebraic manipulation is crucial for solving inequalities and equations alike. It involves various operations such as adding, subtracting, multiplying, dividing, and factoring, applied to both sides of an equation or inequality to isolate the variable. The goal is to rearrange the terms to have all variables on one side and all constants on the other.

In our exercise, we initially had \(5 - 7x < 2x - 4\). By subtracting \(2x\) from both sides and adding 4, we consolidated all the x terms on one side and the constants on the other: \(5 - 9x < -4\). Then, we isolated x by dividing both sides by -9 (and remembering to flip the inequality because we divide by a negative), resulting in \(x > \frac{5}{9}\). This final step is crucial for expressing the solution in terms of the variable, which can then be translated into interval notation for a complete representation of the solution set.