Problem 48 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |x+4|<8 $$

Step-by-Step Solution

Answer

The solution is \(-12 < x < 4\) or \((-12, 4)\).

Step 1: Understand the Inequality

The inequality given is \(|x+4|<8\). This represents all values of \(x\) for which the absolute value of \(x+4\) is less than 8.

Step 2: Set Up the Double Inequality

The absolute value inequality \(|x + 4| < 8\) can be rewritten as a double inequality: \(-8 < x + 4 < 8\).

Step 3: Isolate the Variable

Isolate \(x\) by subtracting 4 from all parts of the inequality: \(-8 - 4 < x + 4 - 4 < 8 - 4\), which simplifies to \(-12 < x < 4\).

Step 4: Write the Solution in Interval Notation

The solution set in interval notation is \((-12, 4)\).

Step 5: Graph the Solution Set

Graph the solution set on a number line by drawing an open interval from -12 to 4. Use open circles at -12 and 4 to indicate these endpoints are not included in the interval.

Key Concepts

Interval Notation

Interval notation is a way of representing a range of values on the real number line. It is particularly useful in solving inequalities. To use interval notation, we follow these rules:

• Use round brackets \( ... \) to indicate that an endpoint is not included (called an open interval).
• Use square brackets \[ ... \] to indicate that an endpoint is included (called a closed interval).

In our solved exercise, we found that the solution set for the inequality \(|x + 4| < 8\) is \(-12 < x < 4\). This translates to an open interval in interval notation as \((-12, 4)\). Since neither endpoint is included, we use round brackets.

Graphing Inequalities

When graphing inequalities on a number line, we visualize the solution set by shading the regions where the inequality holds true. For our current example, we are graphing \(-12 < x < 4\). Here's what we do step-by-step:

• Identify the boundary points from the inequality. Here, they are -12 and 4.
• Draw open circles around -12 and 4 because these points are not included in the solution set. Open circles represent values that are not part of the solution.
• Shade the region between -12 and 4, since these are the values of x that satisfy the inequality. Shading indicates that all numbers within this range are solutions to the inequality.

Double Inequality

A double inequality is an expression in which a variable is constrained within two bounds. It is written in the form \(a < x < b\). This type of inequality means that \x\ must simultaneously satisfy both inequalities. In our example \(|x + 4| < 8\), we converted it into a double inequality: \(-8 < x + 4 < 8\). To solve this:

• Subtract 4 from every part: \-8 - 4 < x + 4 - 4 < 8 - 4\.
• Which simplifies to: \-12 < x < 4\.

Thus, \x\ is constrained to lie between -12 and 4. The overlapping parts of these two simple inequalities give us the solution set.