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Interval notation is a method for writing down the set of all numbers between a starting point and an endpoint. This notation is concise and gives a clear visual presentation of the solution to inequalities. In the case of the inequality provided, after solving for \(x\), we have \(x > -2\). This means x can be any number greater than -2, but not including -2 itself.

When using interval notation:

• \((a, b)\) means all numbers between \(a\) and \(b\), but excluding \(a\) and \(b\) themselves.
• \([a, b)\) means all numbers from \(a\) to \(b\), including \(a\), but excluding \(b\).
• \((a, b]\) means all numbers from \(a\) to \(b\), excluding \(a\), but including \(b\).
• \([a, b]\) means all numbers from \(a\) to \(b\), including both \(a\) and \(b\).

In the example \((-2, \infty)\), parentheses around -2 show that -2 is not part of the solution. The infinity symbol \(\infty\) always has a parenthesis around it because infinity is not a number, but a concept, and cannot be "equaled" or included. Thus, \((-2, \infty)\) means all numbers greater than -2.

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality signs such as \(<\), \(>\), \(\leq\), or \(\geq\). Solving them involves finding the range of values that satisfy the inequality.

Consider the inequality \(3 - 5x < 13\). The inequality means we need to find values of \(x\) that will make this expression true.

• First, isolate the terms involving \(x\) by subtracting constants from both sides.
• Then, divide or multiply all terms as needed to solve for \(x\). But remember that multiplying or dividing both sides by a negative number flips the inequality sign.

In any solution process, checking the steps is crucial. These steps can determine if you've correctly flipped the inequality sign. Inequalities allow us to see a wide range of possible solutions, which is helpful for real-world problems where multiple solutions might be acceptable.

Solving inequalities often involves using algebraic manipulation similar to solving equations. These steps are a systematic approach to isolating the variable of interest. In our particular inequality, we performed the following algebraic steps:

**Subtract to Isolate:** Start by subtracting 3 from both sides to move the constant term away from the \(x\) term. This is the initial step in isolating the variable. The result of this step was \(-5x < 10\).

**Divide to Solve:** The next step was to divide by the coefficient of \(x\), which was \(-5\). Dividing across an inequality by a negative number reverses the inequality sign, yielding \(x > -2\).

These steps highlight the importance of operations balancing both sides of the inequality while considering sign changes. By carefully performing these steps, the correct solution confines our variable to the path dictated by the inequality.