Problem 69 - College Algebra

Question

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|x-1| \geq 2$$

Step-by-Step Solution

Answer

The solution for the inequality |x - 1| >= 2 is x ≥ 3 or x ≤ -1. In interval notation, the solution is (-∞, -1] ∪ [3, ∞).

Step 1: Interpret Absolute Value Inequality Without Bars

Interpret the inequality |x - 1| >= 2 as two separate inequalities, x - 1 >= 2 and -(x - 1) >= 2.

Step 2: Solve Each Inequality

Simplify each inequality to obtain x >= 3 and x <= -1.

Step 3: Combine The Results

The solution set for the original inequality is the union of the solution sets for the two inequalities. This means x is greater than or equal to 3 or less than or equal to -1.

Step 4: Express The Solution on Number Line and Interval Notation

In interval notation, the solution set is represented as (-∞, -1] ∪ [3, ∞). On the number line, mark -1 and 3 with closed circles (because the inequality includes equal to) and shade to the left of -1 and to the right of 3.

Key Concepts

Inequality Solving

Solving absolute value inequalities involves a few crucial steps that make the problem more manageable. The first step is to understand what an absolute value inequality expresses. For example, let's consider the inequality \(|x-1| \geq 2\). The absolute value tells us that the distance from \(x\) to 1 must be at least 2 units on the number line.

To remove the absolute value bars, we need to break this inequality into two separate cases:

• The first case is \(x - 1 \geq 2\), which implies the distance is in the positive direction.
• The second case is \(-(x - 1) \geq 2\), implying the distance is in the negative direction.

When we simplify these inequalities, we end up with \(x \geq 3\) and \(x \leq -1\).

The overall solution to \(|x-1| \geq 2\) is the union of these results, meaning any number that satisfies either inequality works. It's important to note how we interpret the symbol "\( \geq\)" which means "greater than or equal to". This ensures that we include the endpoints when expressing the solution.

Interval Notation

Interval notation is a compact and efficient way to describe a set of numbers on the number line. This notation uses brackets and parentheses to show intervals of numbers that are part of a solution set.

For the solution \(x \geq 3\) or \(x \leq -1\) from our inequality problem, we express this using two intervals:

• \(( -\infty, -1 ]\)
• \([ 3, \infty )\)

The notation \(( -\infty, -1 ]\) means that it includes all numbers less than or equal to -1, extending to negative infinity. This is indicated by the closed bracket "]" at -1, showing that -1 is included in the solution.

Similarly, the interval \([ 3, \infty )\) includes all numbers greater than or equal to 3, extending to positive infinity. The closed bracket "[" at 3 shows that 3 is part of the solution set. The infinite intervals are always open, because infinity is not a fixed number.

Graphing Inequalities

Graphing inequalities on a number line visually represents which numbers satisfy the inequality. For example, to graph the solutions for \(x \geq 3\) and \(x \leq -1\):

1. Begin by drawing a horizontal number line.2. Mark the points \(-1\) and \(3\) on this line.3. For the inequality \(x \leq -1\), shade everything to the left of \(-1\). Use a closed circle at \(-1\) because \(-1\) is included in the solution.
4. Similarly, for \(x \geq 3\), shade the number line to the right of \(3\) with a closed circle on \(3\), indicating 3 is part of the solution.

These shading techniques help clearly show all the values that satisfy either inequality. Graphing provides a visual confirmation that our solution spans the intervals \(( -\infty, -1 ]\) and \([ 3, \infty )\), covering all numbers shaded on the number line.