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Interval notation is a concise way to represent a range of numbers, especially when dealing with inequalities. It highlights the set of all possible solutions in one simple expression. - **Brackets and Parentheses**: Brackets \[ ..., ... \] are used to include endpoints in an interval, while parentheses \( ..., ... \) exclude them. For example, \[0, 5\] means all numbers from 0 to 5, including 0 and 5. However, \(0, 5\) means numbers between 0 and 5, excluding the endpoints.- **Infinites and Unbounded Intervals**: When an interval extends infinitely in one direction, use \( \infty \) or \( -\infty \). Always use parentheses with infinity as it is not a number that can be reached or included. For instance, \((-\infty, -1]\) includes all numbers less than or equal to \(-1\), but stretches to infinity.- **Reading the Solution**: In our example, solving the inequality gives \( x \leq -1 \), translating to the interval \((-\infty, -1]\). This interval contains every number up to and including \(-1\), covering all values that \(x\) can take.

A number line graph is a visual representation that helps to easily see the range of solutions to an inequality. It provides an immediate comprehension of where all possible solutions lie.- **Basic Layout**: A number line is a horizontal line marked with numbers in order of increasing value. Each point on this line corresponds to a number.- **Shading and Symbols**: For an inequality like \(x \leq -1\), draw a line extending from \(-\infty\) to \(-1\). Shade this line to show all included numbers. At \(-1\), place a closed dot to signify that \(-1\) is part of the solution set.- **Importance and Usage**: Using the number line makes it straightforward to interpret and communicate the extent of solutions. For students, it's a helpful tool to check their solution visually after solving the inequality.

Solving inequalities is crucial for finding possible values that satisfy conditions set by an inequality. Here's a step-by-step breakdown:- **Simplification**: Start by simplifying the inequality. Distribute any factors, and combine like terms on both sides. In our inequality \(4 - 3x \leq -(1 + 8x)\), distribute the negative to simplify to \(4 - 3x \leq -1 - 8x\).- **Balancing both sides**: Add or subtract terms on both sides to isolate terms involving the variable. Add \(8x\) to both sides to cancel it out on the right, simplifying to \(4 + 5x \leq -1\).- **Isolating the Variable**: Get the variable term by itself. Subtract \(4\) from both sides to continue isolating \(x\), which gives \(5x \leq -5\).- **Solving for the Variable**: Finally, divide to solve for the variable. Divide both sides by 5 to find \(x \leq -1\).- **Applying Solution**: With \(x \leq -1\), express it in interval notation as \((-\infty, -1]\). Graph it to visually confirm the solution.Understanding these steps ensures that you can solve any linear inequality with confidence.