Problem 59 - Algebra 1: Concepts and Skills

Question

Solve the absolute-value inequality. (Lesson 6.7) $$|x+13| \geq 33$$

Step-by-Step Solution

Answer

The solution to the inequality \(|x+13| \geq 33\) is \(x \leq -46\) or \(x \geq 20\).

Step 1: Identify the two cases

The first step to solve \(|x+13| \geq 33\) is to set up two separate inequalities based on the definition of the absolute-value inequality. These cases are \(x+13 \geq 33\) (the positive case) and \(-(x+13) \geq 33\) (the negative case).

Step 2: Solve the positive case

To solve the positive case, \(x+13 \geq 33\), subtract 13 from both sides to isolate \(x\), resulting in \(x \geq 20\).

Step 3: Solve the negative case

To solve the negative case, \(-(x+13) \geq 33\), multiply the inequality by -1 to make it \((x+13) \leq -33\), which simplifies to \(x \leq -46\) after subtracting 13 from both sides.

Step 4: Combine the solutions

These solutions combine to give one solution consisting of two ranges. The solution to the original absolute-value inequality \(|x+13| \geq 33\) is \(x \leq -46\) or \(x \geq 20\).

Key Concepts

Positive Case

An absolute-value expression, such as \(|x+13| \geq 33\), considers two scenarios. The first is the positive case. Here, we assume that the expression inside the absolute value is non-negative. In other words, the expression is simply as it is without any changes. For the positive case, the inequality becomes:

• \(x+13 \geq 33\)

Solving this form is straightforward. We perform basic operations to isolate the variable \(x\). First, we subtract 13 from both sides, which simplifies our inequality:

• \(x \geq 20\)

This result tells us that when solving the positive case, \(x\) must be at least 20. The positive case gives us one part of the entire solution for the absolute value inequality.

Negative Case

The negative case considers when the expression inside of the absolute value could be negative. This means the absolute value flips the sign of the expressions to make it positive. For our inequality,

• \(-(x+13) \geq 33\)

This step involves considering the expression \(x+13\) as a negative number. To solve, we first multiply the entire inequality by \(-1\), ensuring to reverse the inequality sign:

• \(x+13 \leq -33\)

Next, solve for \(x\) by subtracting 13 from each side, leading to:

• \(x \leq -46\)

This tells us \(x\) must be less than or equal to \(-46\). The negative case solution reflects the possibility that the expression inside the absolute value could have initially been negative.

Inequality Solutions

After determining the results of both the positive and negative cases, we combine them to comprehend the full solution of the absolute-value inequality. These kinds of inequalities generally provide two possible ranges for solutions, which might sometimes seem "spread apart."For \(|x+13| \geq 33\), our combined solution states:

• \(x \geq 20\)
• or \(x \leq -46\)

These solutions illustrate that \(x\) can be any number greater than or equal to 20, or any number less than or equal to \(-46\). The absolute value inequality presents solutions in this manner because it describes numbers whose distance from \(-13\) is at least 33 units.Understanding the solution means visualizing two separate intervals on the number line which aren't adjacent to each other. This approach is essential when interpreting the result of an absolute value inequality.