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Interval notation is a way to describe sets of numbers along a number line, using specific symbols to indicate whether the endpoints are included or not. For instance, brackets "\([ ]\)" indicate that a number is included in the interval, while parentheses "\(( )\)" suggest it is not.

In the inequality solution from our exercise, you finished with the inequality \(x \geq -2\). This means that \(x\) can be any number starting from \(-2\) and moving toward infinity. To express this as interval notation, we write \([-2, \infty)\). Note the closed bracket at \(-2\) because \(-2\) is part of the solution, while infinity is always accompanied by a parenthesis as it is not a fixed number.

Understanding interval notation helps you clearly and precisely convey solution sets for inequalities.

Solving inequalities is similar to solving equations but with a few extra rules. The main concept is to isolate the variable on one side of the inequality sign \(\leq, \geq, <, \) or \( > \).

1. Begin by simplifying both sides, if needed, to isolate terms.2. When you multiply or divide both sides of an inequality by a negative number, remember this crucial rule: flip the inequality sign. So, \(\leq\) becomes \(\geq\), and \(<\) becomes \(>\), and vice versa.

In our problem, starting from \(-3x - 1 \leq 5\), you first added 1 to clear the constant term, resulting in \(-3x \leq 6\). Dividing by \(-3\) both simplifies \(x\) and flips the inequality, resulting in \(x \geq -2\).

Always perform a check by plugging values back into the original inequality to ensure accuracy.

Graphing inequalities visually represents all possible solutions on a number line. This approach helps you see, at a glance, which values satisfy the inequality.

To graph \(x \geq -2\):

• First, locate \(-2\) on the number line.
• Place a closed dot at \(-2\) because the inequality sign "\(\geq\)" means \(-2\) is included in the solution.
• Draw an arrow extending to the right from \(-2\) to indicate all numbers greater than \(-2\) are included.

This representation matches the interval notation \([-2, \infty)\). By doing this, you communicate the comprehensive set of values that satisfy the original inequality.