Problem 71 - College Algebra

Question

Solve absolute value inequality. \(|x-1| \geq 2\)

Step-by-Step Solution

Answer

The solution to the inequality \(|x - 1| \geq 2\) is \([-∞, -1] \cup [3, ∞]\)

Step 1: Express the absolute value inequality as two separate inequalities

The inequality \(|x - 1| \geq 2\) represents two cases - (1) when \(x - 1\) is greater than or equal to 2, and (2) when \(-(x - 1)\) is greater than or equal to 2. Express these two cases as separate inequalities: Case 1: \(x - 1 \geq 2\)Case 2: \(-(x - 1) \geq 2\)

Step 2: Resolve the inequalities

Now, solve each inequality to find the possible range of \(x\). For Case 1: \(x - 1 \geq 2\), add 1 to both sides of the inequality to get: \(x \geq 3\).Then, for Case 2: \(-(x - 1) \geq 2\), distribute the '-' sign and you'll get: \(-x + 1 \geq 2\). Subtract 1 from both sides to get: \(-x \geq 1\), multiply each side by -1 (and remember to flip the inequality symbol) and you get: \(x \leq -1\)

Step 3: State the solution

The solution to the inequality \(|x - 1| \geq 2\) can now be stated from the solutions of the two conditions: \(x\) is any number that is less than or equal to -1, or greater than or equal to 3. In interval notation, this is written as \([-∞, -1] \cup [3, ∞]\). Note the use of square brackets, they indicate that -1 and 3 are included in the solution set

Key Concepts

Interval Notation

Interval notation is an elegant and concise way to express the set of solutions for inequalities. It uses intervals to represent all the numbers that satisfy an inequality as part of a solution set.
Let's take the solution from our example: We have two separate ranges of numbers that will solve the inequality: one where \(x \leq -1\) and another where \(x \geq 3\).

• The interval \([-∞, -1]\) represents all numbers from negative infinity up to -1. The square bracket indicates that -1 is included in the solution.
• The interval \([3, ∞]\) captures all numbers from 3 to positive infinity. Again, the square bracket shows that 3 is part of the solution.

The union symbol \(\cup\) is used between intervals to combine them, meaning we include numbers from either range. These combined intervals cover all possible solutions for the imposed inequality condition.

Inequality Solutions

Solving inequalities involves finding which values satisfy the inequality. When working with absolute value inequalities, we deal with expressions that have two possible conditions to check.
In our exercise, the absolute value inequality \(|x - 1| \geq 2\) requires us to consider that the absolute value signifies a distance on a number line. Hence:

• The first condition \(x - 1 \geq 2\) considers values where the expression within the absolute value itself is greater than or equal to 2.
• The second condition \(-(x - 1) \geq 2\) reflects situations where the negative of the expression within the absolute value is greater than or equal to 2.

By solving each of these separate inequalities, we arrive at two solution ranges for \(x\). Understanding these resulting inequalities lets us know where a value of \(x\), when fed into the original inequality, would hold true.

Algebraic Manipulation

Algebraic manipulation is critical when resolving inequalities, especially those involving absolute values. It involves rearranging and simplifying expressions to isolate the variable.
Let's explore this through our example: In Case 1 \(x - 1 \geq 2\), we add 1 to both sides to solve for \(x\), yielding \(x \geq 3\).
For Case 2 \(-(x - 1) \geq 2\), careful handling is necessary:

• Distribute the negative sign resulting in \(-x + 1 \geq 2\).
• Subtract 1 from both sides leading to \(-x \geq 1\).
• Multiply both sides by -1, and remember to flip the inequality sign, giving \(x \leq -1\).

Each step requires careful execution to correctly follow algebraic rules. Indeed, flipping the inequality sign when multiplying by a negative number is a key operation that affects the direction of the solution.