Problem 23 - College Algebra with Corequisite Support

Question

For the following exercises, solve the inequality involving absolute value. Write your final answer in interval notation. $$ \left|\frac{x-3}{4}\right|<2 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((-5, 11)\).

Step 1: Understanding the Absolute Value Inequality

The inequality involves an absolute value. The expression \( \left| \frac{x-3}{4} \right| < 2 \) means the distance between \( \frac{x-3}{4} \) and 0 is less than 2. This can be separated into two inequalities: \( -2 < \frac{x-3}{4} < 2 \).

Step 2: Solving the Lower Bound Inequality

Start solving the inequality \( -2 < \frac{x-3}{4} \). To eliminate the fraction, multiply every part of the inequality by 4: \( -8 < x - 3 \). Then, add 3 to each part to isolate \( x \): \( -5 < x \).

Step 3: Solving the Upper Bound Inequality

Now solve \( \frac{x-3}{4} < 2 \). Again, multiply every part of the inequality by 4: \( x-3 < 8 \). Then, add 3 to both sides to isolate \( x \): \( x < 11 \).

Step 4: Combining the Results

The solution to the inequality \( \left| \frac{x-3}{4} \right| < 2 \) is obtained by combining the results from the two inequalities: \( -5 < x < 11 \).

Step 5: Writing the Solution in Interval Notation

In interval notation, the solution \( -5 < x < 11 \) is written as \((-5, 11)\). This represents all the numbers between -5 and 11, not including -5 and 11 themselves.

Key Concepts

Interval Notation

Interval notation is a way of describing the set of solutions for an inequality on a number line. Instead of listing every possible number that fits the inequality, interval notation gives a concise representation. For example, the solution to an inequality like \(-5 < x < 11\) can be indicated as \((-5, 11)\). This means that \(x\) is any number greater than \(-5\) and less than \(11\).

• The parentheses \(( )\) are used to denote that the endpoints are not included in the solution set.
• If the endpoints were included, square brackets \([ ]\) would be used instead.
• For instance, the notation \([-5, 11)\) would include the number \(-5\) but still exclude \(11\).

This method of writing solutions makes expressions easier to read and can help you visually picture the solution set on the number line. Let's consider solving absolute value inequalities using this approach.

Solving Linear Inequalities

To solve linear inequalities, you follow similar steps to solving regular linear equations, but with a few differences. The given problem \(\left|\frac{x-3}{4}\right| < 2\) involves an absolute value, which adds extra steps.

• Start by understanding that the absolute value sees how far a number is from zero on the number line, meaning it is always non-negative.
• An inequality like this can be split into two separate inequalities without the absolute value. For example, \(\left|a\right| < b\) becomes \(-b < a < b\).
• With our case, it required splitting into \(-2 < \frac{x-3}{4} < 2\).

Next, tackle each piece of the split inequality separately:

• For \(-2 < \frac{x-3}{4}\), multiply through by \(4\) to eliminate the fraction, leading to \(-8 < x - 3\), then add \(3\) to solve for \(x\), resulting in \(-5 < x\).
• For \(\frac{x-3}{4} < 2\), follow a similar process to \(x < 11\).

Finally, combine the results to form a compound inequality: \(-5 < x < 11\). This represents the range of \(x\) that satisfies the original inequality.

Understanding Inequalities

Inequalities are like equations, but instead of stating that two expressions are equal, they show a relationship where one is larger or smaller than the other. There are several key symbols used:

• \(<\) means "less than."
• \(>\) means "greater than."
• \(\leq\) or \(\geq\) include the value itself, meaning "less than or equal to" or "greater than or equal to."

Understanding inequalities involves recognizing the direction and meaning they convey:

• When inequalities involve absolute values, it signals not just a single range but potentially two opposite directions on the number line.
• For absolute values within inequalities, consider both the positive and negative range of solutions.

This dual approach to inequalities becomes crucial when dealing with expressions like absolute value because the distance from zero could be either side of the axis.
Lastly, it's important to always check your solution's consistency with the original inequality after solving it, as sometimes mistakes can happen if terms or operations are overlooked during the solving process. This reinforces your understanding of the relationship described by the inequality.