Problem 64 - Algebra 2

Question

Solve each inequality. \(|x+4|>2\)

Step-by-Step Solution

Answer

The solution is \(x < -6\) or \(x > -2\).

Step 1: Understand the Absolute Value Inequality

The inequality \(|x+4| > 2\) means that the expression inside the absolute value, which is \(x+4\), is more than 2 units away from zero on the number line. This creates two separate inequalities to solve.

Step 2: Define Two Separate Inequalities

To solve \(|x+4| > 2\), we need to split it into two separate cases: \(x+4 > 2\) and \(x+4 < -2\). These represent the distances from zero being more than 2.

Step 3: Solve the First Inequality

Solve the inequality \(x+4 > 2\). Subtract 4 from both sides: \[x > 2 - 4\] resulting in \(x > -2\).

Step 4: Solve the Second Inequality

Solve the inequality \(x+4 < -2\). Subtract 4 from both sides: \[x < -2 - 4\] giving \(x < -6\).

Step 5: Combine Solutions

The solutions to the inequalities \(x > -2\) and \(x < -6\) are combined by using the word 'or' to show they are separate solution sets. Therefore, the solution to the original inequality is \(x < -6\) or \(x > -2\).

Step 6: Solution in Interval Notation

In interval notation, the solution \(x < -6\) or \(x > -2\) is represented as \((-fty, -6) \cup (-2, fty)\).

Key Concepts

Interval Notation

Interval notation is a way to express the range of values that satisfy an inequality. It provides a concise format that helps in understanding which values are included in a solution set. For our inequality, the solution is split into two parts: - One part includes all values less than -6. - The other part includes all values greater than -2. By representing these solutions in interval notation, we write them as:

• For values less than -6: (-∞, -6)
• For values greater than -2: (-2, ∞)

The union symbol (∪) connects these two intervals indicating that the complete solution includes any number from either range. It's important to note that parentheses in interval notation denote that the endpoints are not included, which is why we use "(" and ")" rather than "[" and "]". Make sure to always double-check your interval notation by reviewing each part of the solution set.

Inequality Solving Strategies

When dealing with absolute value inequalities like |x+4| > 2, there are strategic steps to consider.

• Understand that the expression inside the absolute value represents a distance from zero on the number line. This means we need to handle two scenarios because the distance can apply in both directions from zero.
• For |x+4| > 2, we convert it into two distinct inequalities: (x+4 > 2) and (x+4 < -2).
• Solving it involves dealing with these two separate conditions to find all values of x that satisfy either one or both.

These strategies allow us to frame the problem effectively and guide our steps. This way, you don’t miss out on any part of the solution that could satisfy the original inequality.

Algebraic Manipulation

Algebraic manipulation is essential for solving inequalities, especially absolute value ones. It involves transforming equations or inequalities into an easier-to-solve form. Here's how we can apply it:

• First, break down the absolute value inequality, such as |x+4| > 2, into two separate inequalities based on its definition.
• Apply algebraic steps to solve each inequality. For (x+4 > 2), subtract 4 from both sides to isolate x. This results in x > -2.
• Similarly, for (x+4 < -2), perform the same operation: subtract 4, leading to the solution x < -6.

These manipulations allow you to transform the complex form into simpler equivalent equations. It's crucial to handle each step methodically to avoid errors and achieve the correct solution. Practicing these skills will aid in efficiently solving similar problems in the future.