Problem 32 - Algebra and Trigonometry

Question

\(23-48\) Solve the inequality. Express the answer using interval notation. $$ |x+1| \geq 3 $$

Step-by-Step Solution

Answer

Solution: \((-\infty, -4] \cup [2, \infty)\).

Step 1: Understand the Inequality

The given inequality \(|x+1| \geq 3\) involves an absolute value. This means the expression inside the absolute value can be either equal to or greater than 3 or equal to or less than -3.

Step 2: Split the Inequality

Since absolute values create two scenarios, we need to consider both: 1. \(x+1 \geq 3\) 2. \(x+1 \leq -3\)

Step 3: Solve for x in Scenario 1

For \(x+1 \geq 3\): - Subtract 1 from both sides to isolate \(x\). - The resulting inequality is \(x \geq 2\).

Step 4: Solve for x in Scenario 2

For \(x+1 \leq -3\): - Subtract 1 from both sides to isolate \(x\). - The resulting inequality is \(x \leq -4\).

Step 5: Write the Solution in Interval Notation

The solutions from both scenarios are combined to express the overall solution as two intervals: - \(x \geq 2\) corresponds to the interval \([2, \infty)\).- \(x \leq -4\) corresponds to the interval \((-\infty, -4]\).Thus, in interval notation, the solution is \((-\infty, -4] \cup [2, \infty)\).

Key Concepts

absolute value inequalities

Absolute value inequalities are problems that involve the absolute value of a variable expression. The absolute value of a number represents its distance from zero on a number line, regardless of direction. Therefore, the absolute expression

• must be non-negative
• indicates that the value can either be on the positive end or the negative end of the number line, at an equal distance from zero

When solving absolute value inequalities like \[|x+1| \geq 3\],we have to consider two cases due to the nature of absolute values.
In this example,

• The inequality \[x+1 \geq 3\] considers the scenario where the expression \(x+1\) stays to the right of 3 on the number line.
• While \[x+1 \leq -3\] considers the scenario where it moves to the left of -3 on the number line.

Both cases are viable because the absolute value only determines the distance, not the direction.Understanding this helps to split and solve the inequality separately to find all possible solutions.

interval notation

Interval notation is a way of representing a set of numbers between two endpoints on the number line. It is often used in the solutions of inequalities to clearly show the range of values that satisfy a given condition. Symbols are employed to define closed or open intervals.

• Closed interval: Includes the endpoint, represented by square brackets, for example, \[[2, 5]\].
• Open interval: Excludes the endpoint, denoted by parentheses, like \((2, 5)\).
• Infinite interval: When the set extends infinitely in one or both directions, we use infinity symbols, such as \((-\infty, 3]\) or \([3, \infty)\).

For our inequality solution, \((-\infty, -4] \cup [2, \infty)\),the interval notation signifies all numbers less than or equal to -4, or greater than or equal to 2.
This notation succinctly and precisely captures both parts of a compound inequality.

compound inequalities

Compound inequalities involve two separate inequalities that are linked by the word 'and' or 'or'. These links determine whether we are finding the overlap of solutions (intersection) or the collection of both possibilities (union).

• 'And' implies both conditions must be true at the same time, such as \(x > 1\) and \(x < 5\), which is typically represented as\((1, 5)\).
• 'Or' means that solutions valid under either condition are acceptable. For instance, \(x \leq -4\) or \(x \geq 2\).In interval notation, this can look like \((-\infty, -4] \cup [2, \infty)\),where solutions fall into either one of the intervals.

In our example, solving \(|x+1| \geq 3\)produces a compound inequality because it leads to two sets of solutions combined with an 'or'.
Thus, compound inequalities allow us to represent complex, multi-part solutions that cover various portions of the number line.