Problem 25 - Algebra and Trigonometry

Question

Solve the inequality. Express the answer using interval notation. $$ |x+1| \geq 1 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -2] \cup [0, \infty)\).

Step 1: Understand Absolute Value Inequality

An absolute value inequality of the form \(|x+a| \geq b\) can be rewritten as two separate inequalities: \(x+a \geq b\) or \(x+a \leq -b\). Our task is to solve for \(x\) in these two cases.

Step 2: Rewrite Using Two Cases

Given the inequality \(|x+1| \geq 1\), rewrite it as two separate cases: 1. \(x+1 \geq 1\)2. \(x+1 \leq -1\)

Step 3: Solve First Inequality

Solve \(x+1 \geq 1\): Subtract 1 from both sides to isolate \(x\).\[x + 1 - 1 \geq 1 - 1\]\[x \geq 0\]

Step 4: Solve Second Inequality

Solve \(x+1 \leq -1\): Subtract 1 from both sides to isolate \(x\).\[x + 1 - 1 \leq -1 - 1\]\[x \leq -2\]

Step 5: Combine Results Using Union

The solutions from both inequalities are combined using a union because the absolute value inequality uses "greater than or equal to." Thus, the solution is the set of all \(x\) such that \(x \geq 0\) or \(x \leq -2\).

Step 6: Express Solution in Interval Notation

Express the combined solution in interval notation. From the solutions of the two inequalities:- \(x \geq 0\) corresponds to the interval \([0, \infty)\).- \(x \leq -2\) corresponds to the interval \((-\infty, -2]\).Combine them to form the final answer: \[(-\infty, -2] \cup [0, \infty)\]

Key Concepts

Interval Notation

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.

Inequality Solution

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.

Algebraic Expression

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.