Problem 29 - Algebra and Trigonometry

Question

\(23-48\) 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: Understanding the Absolute Value Inequality

The inequality given is \(|x+1| \geq 1\). An absolute value inequality of the form \(|x - a| \geq b\) implies that the expression inside has two scenarios: it is either greater than or equal to \(b\) or less than or equal to \(-b\). So, we split the inequality into two separate inequalities: \(x+1 \geq 1\) or \(x+1 \leq -1\).

Step 2: Solve the First Inequality

Let's solve the first inequality: \(x+1 \geq 1\). Subtract 1 from both sides to isolate \(x\). This gives \(x \geq 0\).

Step 3: Solve the Second Inequality

Now, solve the second inequality: \(x+1 \leq -1\). Again, subtract 1 from both sides to isolate \(x\). This gives \(x \leq -2\).

Step 4: Combine Solutions

The two solutions, \(x \geq 0\) and \(x \leq -2\), can be combined since \(|x+1| \geq 1\) implies either condition must hold true. Therefore, the solution to the inequality is \(x \in (-\infty, -2] \cup [0, \infty)\).

Step 5: Express in Interval Notation

Given the combinations of the solutions from both inequalities, we express these as \((-\infty, -2] \cup [0, \infty)\). This represents all values of \(x\) that satisfy the original inequality.

Key Concepts

Interval Notation

When dealing with inequalities, interval notation is a concise way of expressing the range of solutions. This method uses intervals to show us where the solutions reside on a number line. In our case, with the inequality \(|x+1| \geq 1\), we found two distinct intervals:

• \((-\infty, -2]\)
• \([0, \infty)\)

These intervals tell us all the possible values that \(x\) can take to satisfy the inequality. The square brackets \([-2]\) and \([0]\) indicate that the endpoints -2 and 0 are included in the solution. This is because our inequality operator is \(\geq\), which means 'greater than or equal to'. The round parentheses \((-\infty\) and \(\infty)\) show that negative and positive infinity are not inclusive - since infinity is not a number that can be reached, it is never included. Using interval notation is advantageous for quickly communicating a solution set and understanding its position on the number line.

Solving Inequalities

Solving inequalities is about finding the values that make the inequality true. Unlike solving equations, these solutions often involve ranges of numbers rather than specific values. In the example \(|x+1| \geq 1\), the solution process involved multiple steps.

Firstly, we recognized that the absolute value inequality can be split into two different inequalities. This is because the absolute value measures the distance from zero on the number line, so \(|x+1| \geq 1\) implies that

• \(x+1 \geq 1\) or
• \(x+1 \leq -1\)

The next step involved solving each inequality separately. This required basic algebraic manipulation, such as subtracting 1 from both sides:

• For \(x + 1 \geq 1\), subtracting 1 gives us \(x \geq 0\).
• For \(x + 1 \leq -1\), subtracting 1 gives us \(x \leq -2\).

The final step combined these solutions, since the original inequality \(|x+1| \geq 1\) is satisfied if either of the simplified inequalities is true. This process of breaking down and solving inequalities helps us determine all possible numbers that satisfy the condition.

Absolute Value Functions

An absolute value function is a type of function that measures the distance of a number from zero on a number line, always yielding a non-negative result. For instance, \(|x+1|\) measures how far \(x+1\) is from zero, irrespective of its direction. Such functions are characterized by a 'V' shape when graphed.

When working with absolute value inequalities like \(|x+1| \geq 1\), it is essential to understand their split nature due to the non-negative requirement. This inequality means that the expression \(x+1\) must be at least 1 unit away from zero, either on the positive side or the negative side of the number line.

• If \(x+1\) is on the positive side, we derive the inequality \(x+1 \geq 1\).
• If \(x+1\) is on the negative side, the applicable inequality becomes \(x+1 \leq -1\).

The ability of absolute value functions to express distances contributes to solving complex logical conditions within inequalities. Understanding their characteristics helps in visualizing the solution on a number line and confirming the validity of the obtained interval notation.