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Interval notation is a shorthand way of expressing a range of numbers along the number line. It's a critical skill in mathematics and helps in communicating solutions to inequalities efficiently.

In interval notation:

• Brackets [ ] are used when the end number is included in the set, known as a closed interval.
• Parentheses ( ) are used when the end number is not included, known as an open interval.

For example, the interval \( [0, \infty) \) includes all numbers greater than or equal to 0, and the interval \( (-\infty, -2] \) includes all numbers less than or equal to -2.

When using interval notation to express the solution of our inequality \( |x+1| \geq 1 \), we see that the solution consists of two parts:

• One interval for \( x \geq 0 \), written as \( [0, \infty) \)
• Another interval for \( x \leq -2 \), written as \( (-\infty, -2] \)

Combining these intervals with a union symbol (\( \cup \)), our complete solution in interval notation is \( (-\infty, -2] \cup [0, \infty) \). This indicates that any number in these intervals satisfies the given inequality.

Solving inequalities involves finding the range of values that satisfy a given condition. When we deal with absolute value inequalities like \( |x+1| \geq 1 \), we're looking for values of \( x \) that make this expression true.

The absolute value expression \( |x+1| \) can represent two scenarios:

• When the expression inside the absolute value is non-negative, i.e., \( x+1 \geq 0 \).
• When the expression is negative, i.e., \( x+1 < 0 \).

For the inequality \( |x+1| \geq 1 \), we consider two separate cases:

• Case 1: \( x+1 \geq 1 \), which simplifies to \( x \geq 0 \).
• Case 2: \( x+1 \leq -1 \), which simplifies to \( x \leq -2 \).

Combining the solutions from these cases, the complete solution set is expressed in interval notation as \( (-\infty, -2] \cup [0, \infty) \). This means that any value in these sets will satisfy the original inequality.

An algebraic expression is a combination of numbers, symbols, and operators (like +, -, \, *, etc.) that represent a specific value.

The expression within our inequality, \( x+1 \), is an example of a simple algebraic expression. It represents the values that \( x \) can take when modified by the addition of 1.

Algebraic expressions are fundamental in solving inequalities because they:

• Allow us to manipulate and simplify equations and inequalities.
• Help translate word problems into mathematical statements.
• Provide a method to evaluate different scenarios through substitution.

In the context of our inequality \( |x+1| \geq 1 \), understanding how to manipulate the algebraic expression \( x+1 \) is crucial.

To solve, we split the absolute value expression into two scenarios:

• Firstly, considering if \( x+1 \) is greater than or equal to the right side, leading to \( x \geq 0 \).
• Secondly, considering if \( -(x+1) \) is less than or equal to the right side, leading to \( x \leq -2 \).

These transformations demonstrate how algebraic expressions lie at the heart of formulating and solving inequalities.