Problem 35 - College Algebra

Question

For the following exercises, graph the function. Observe the points of intersection and shade the \(x\) -axis representing the solution set to the inequality. Show your graph and write your final answer in interval notation. $$ |x+7| \leq 4 $$

Step-by-Step Solution

Answer

The solution set is \([-11, -3]\) in interval notation.

Step 1: Understanding the Absolute Value Inequality

The given inequality is \(|x + 7| \leq 4\). An absolute value inequality \(|a| \leq b\) means the expression \(a\) must be between \(-b\) and \(b\). Here, \[-4 \leq x + 7 \leq 4\].

Step 2: Solving the Compound Inequality

We will solve the two inequalities separately: 1. \(-4 \leq x + 7\) 2. \(x + 7 \leq 4\). For the first inequality: \(-4 - 7 \leq x \) \(-11 \leq x\). For the second inequality: \(x \leq 4 - 7\) \(x \leq -3\).

Step 3: Write the Combined Solution

The solution for \(|x + 7| \leq 4\) is the intersection of the two inequalities we solved. Thus, the solution is \[-11 \leq x \leq -3\].

Step 4: Graph the Inequality

On a number line, plot the points \(-11\) and \(-3\). Use solid dots as both are inclusive. Shade the region between these points, as this represents values of \(x\) that satisfy the inequality.

Step 5: Express in Interval Notation

The shaded region on the number line corresponds to the interval \([-11, -3]\). This interval notation indicates that \(x\) includes all real numbers between \(-11\) and \(-3\), inclusive.

Key Concepts

Compound Inequalities

Compound inequalities involve expressions that include more than one inequality. In the context of absolute value inequalities, such as \( |x+7| \leq 4 \), it implies the expression inside the absolute value brackets must be simultaneously less than or equal to the positive number, and greater than or equal to the negative of that number. In our exercise, the absolute value inequality translates to a compound inequality: \( -4 \leq x + 7 \leq 4 \). This inequality is termed 'compound' because it "compounds" the possible solutions into two ranges that meet specific criteria. To solve this compound inequality:

• Solve \( -4 \leq x + 7 \) by isolating \( x \), leading to \( x \geq -11 \).
• Solve \( x + 7 \leq 4 \) by further isolating \( x \), resulting in \( x \leq -3 \).

The solution to the inequality is found where these solutions overlap, indicating that \( x \) can take any value between \( -11 \) and \( -3 \), inclusive.

Interval Notation

Interval notation is a way to describe a set of numbers along the number line. It is particularly useful in expressing solutions to inequalities. In our case, the inequality solution \( -11 \leq x \leq -3 \) is expressed in interval notation.This is done by including:

• The smallest number of the interval, \( -11 \), and the largest number, \( -3 \).
• Square brackets \( [ ] \) to indicate that the endpoints are included in the solution set, known as "inclusive."

Therefore, the interval notation is \([-11, -3]\) which represents all numbers \( x \) that are equal to or between \( -11 \) and \( -3 \). Interval notation provides a concise and clear way to express a continuous set of numbers, making it very practical for solution sets.

Graphing Inequalities

Graphing inequalities on a number line visually represents the solution set. It provides a clear picture of what numbers satisfy the given condition. In our inequality \( |x+7| \leq 4 \), once resolved into \(-11 \leq x \leq -3\), we use the number line for a graphical representation.To graph this inequality:

• Draw a number line and mark the endpoints \( -11 \) and \( -3 \).
• Use solid dots at these points to indicate they are included as part of the solution (since our inequality is non-strict, using \( \leq \)).
• Shade the entire region between these points. This shaded area represents all possible \( x \) values that satisfy the inequality.

This visual aid helps to understand the extent of the solution set quickly. Graphing is an effective tool for both teaching and learning, making it more straightforward to grasp the concept of inequalities.