Problem 19 - College Algebra

Question

Solve the inequality involving absolute value. Write your final answer in interval notation. $$ |x-2|+4 \geq 10 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-\infty, -4] \cup [8, \infty)\).

Step 1: Isolate the Absolute Value

Begin by isolating the absolute value expression. The given inequality is \(|x-2| + 4 \geq 10\). Subtract 4 from both sides to isolate the absolute value: \(|x-2| \geq 6\).

Step 2: Split the Absolute Value Inequality

The inequality \(|x-2| \geq 6\) implies two separate inequalities because the expression inside the absolute value can either be 6 units greater or 6 units less than 0. This gives us two cases: \(x-2 \geq 6\) or \(x-2 \leq -6\).

Step 3: Solve the First Inequality

Solve the inequality \(x-2 \geq 6\). Add 2 to both sides to find \(x \geq 8\).

Step 4: Solve the Second Inequality

Solve the inequality \(x-2 \leq -6\). Add 2 to both sides to find \(x \leq -4\).

Step 5: Combine the Solutions

Combine the solutions from Step 3 and Step 4. The solution to the inequality is the union of these two conditions, which can be expressed in interval notation as \((-\infty, -4] \cup [8, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way to express a set of numbers along a number line. It is especially useful for expressing solutions of inequalities and highlighting ranges of numbers that meet certain conditions. Learning how to read and write interval notation helps communicate mathematical concepts clearly and concisely.

There are a few components to remember when writing interval notation:

• **Brackets**: Brackets are used to include or exclude boundary points. A square bracket \( [ or ] \) indicates that the endpoint is included, while a round bracket \( ( or ) \) means the endpoint is excluded from the set.
• **Infinity Symbols**: The symbols \( \infty \) and \( -\infty \) are used to denote unbounded ends of the interval, indicating that the set continues indefinitely in that direction.
• **Union**: The union symbol \( \cup \) combines multiple intervals, reflecting that any value in either interval satisfies the condition.

For the exercise \((\-\infty, -4] \cup [8, \infty)\)), it means that the solution includes all values less than or equal to \(-4\) and all values greater than or equal to \(8\). This indicates two separate ranges where the inequality condition holds true.

Solving Inequalities

Solving inequalities involves finding all the possible values for the variable that make the inequality true. The process is similar to solving equations but with some specific caution for inequalities:

• **Directional Change**: If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.
• **Isolation**: As demonstrated in the original solution, start by isolating the variable term, just like isolating in equations. For absolute value inequalities, this involves first isolating the absolute value expression.
• **Splitting**: Absolute value inequalities usually require the inequality to be split into two separate inequalities. This is because the absolute value represents the distance from zero, which can mean both positive and negative directions on the number line.

In the given exercise, by setting \(|x-2| \geq 6\), the absolute value inequality splits into two cases: \(x-2 \geq 6\) or \(x-2 \leq -6\), reflecting the two possibilities for distance from 2 being greater than or equal to 6.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. Understanding and manipulating them is essential to solving inequalities and many other algebraic problems.

• **Variables**: Symbols like \(x\) are used to stand in for unknown quantities. Solving the inequality usually involves finding values for these variables that satisfy the given conditions.
• **Arithmetic Operations**: These include addition, subtraction, multiplication, and division. Knowing how to properly apply these operations is fundamental in solving algebraic expressions within inequalities.
• **Transformation**: To maintain equality or the correct inequality during operations, always perform the same operation to both sides of the equation or inequality.

In the problem \(|x-2|+4 \geq 10\), subtracting 4 from both sides simplifies it to \(|x-2| \geq 6\), making it easier to handle. The key is often to change the expression into a form that is straightforward to solve.