Problem 18 - Algebra 2 and Trigonometry

Question

In \(15-26,\) solve each inequality and write the solution set if the variable is an element of the set of integers. $$ |x-3|<4 $$

Step-by-Step Solution

Answer

The solution set is \{0, 1, 2, 3, 4, 5, 6\}.

Step 1: Understand Absolute Value Inequality

To solve the inequality \(|x-3| < 4\), we need to remember that an inequality of the form \(|A| < B\) means \(-B < A < B\). In this case, \(A\) is \(x - 3\) and \(B\) is \(4\). Therefore, the inequality \(|x-3| < 4\) can be rewritten as two separate inequalities: \(-4 < x - 3 < 4\).

Step 2: Solve the Two-Part Inequality

Now, solve the two-part inequality \(-4 < x - 3 < 4\). This compound inequality can be split into two separate inequalities: 1. \(-4 < x - 3\) 2. \(x - 3 < 4\)

Step 3: Solve the First Inequality

Solve the first inequality, \(-4 < x - 3\). Add \(3\) to both sides to isolate \(x\): \(-4 + 3 < x\) Thus, \(-1 < x\).

Step 4: Solve the Second Inequality

Solve the second inequality, \(x - 3 < 4\). Add \(3\) to both sides to isolate \(x\): \(x < 4 + 3\)Thus, \(x < 7\).

Step 5: Combine Both Inequalities

Combine the results of both inequalities: \(-1 < x < 7\). This means that \(x\) is greater than \(-1\) and less than \(7\).

Step 6: List Integer Solutions

Since \(x\) is an integer, we find all integer values that satisfy the inequality \(-1 < x < 7\). These integer values are: 0, 1, 2, 3, 4, 5, and 6. Thus, the solution set is \{0, 1, 2, 3, 4, 5, 6\}.

Key Concepts

Understanding Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is portrayed using vertical bars, such as in \( |x-3| \). The essence of absolute value is distance, which is always a non-negative number. This means whether a number is positive or negative, its absolute value converts it to a positive form.
For example, \(|-5| = 5\) and \( |5| = 5\), which illustrates that both -5 and 5 are equally distant from zero.
This concept is crucial when dealing with inequalities involving absolute values, as it affects how we interpret and solve them.

Identifying Integer Solutions

Integer solutions are specific solutions of an inequality where the solution must be a whole number, not a fraction or decimal. In our inequality \(|x-3| < 4\), although the general solution is given by the compound inequality \(-1 < x < 7\), we are interested in integer values that fit within this range.
These integers include 0, 1, 2, 3, 4, 5, and 6.

• 0 is the smallest whole number greater than -1.
• 6 is the largest whole number less than 7.

Understanding integer solutions helps confirm which values satisfy the original inequality when restricted to integers only.

Working with Compound Inequalities

Compound inequalities like \(-4 < x-3 < 4\) are composed of two inequalities joined together. In this scenario, they imply a range within which the variable lies.
To solve compound inequalities, they are split into two separate, simpler inequalities.

• First: \(-4 < x-3\) translates to \(-1 < x\)
• Second: \(x-3 < 4\) translates to \(x < 7\)

By addressing both parts individually and then combining the results, we find the solution to the original inequality as a range. This dual approach allows for a clear understanding of possible solutions and helps accurately define the solution set.

Solving Inequalities with Absolute Values

Inequalities with absolute values often frighten because of their non-traditional format. However, they can be approached systematically. The key is rewriting them without absolute value symbols.
This involves recognizing that \(|x-3| < 4\) can be expressed as the compound inequality \(-4 < x-3 < 4\).
This step effectively transforms a seemingly complex absolute value problem into a more straightforward two-part inequality. Once split, it's a matter of solving each part:

• Add 3 to solve \(-4 < x-3\) to get \(-1 < x\).
• Add 3 to solve \(x-3 < 4\) to get \(x < 7\).

Thus, understanding how to handle absolute value inequalities permits you to find solutions in a logical and stress-free way.