Problem 55 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Write the solution in interval notation. $$|3 x-1|<8$$

Step-by-Step Solution

Answer

The solution is \((-\frac{7}{3}, 3)\).

Step 1: Understand the Absolute Value Inequality

The inequality is \(|3x - 1| < 8\). The absolute value \(|a| < b\) implies that the expression inside the absolute value lies between \(-b\) and \(b\). So, we rewrite it as two separate inequalities: \(-8 < 3x - 1 < 8\).

Step 2: Solve Inequality on the Left

Starting with the left part of the compound inequality \(-8 < 3x - 1\), we solve for \(x\). Add 1 to both sides to get \(-7 < 3x\). Then, divide both sides by 3 to isolate \(x\), resulting in \(-\frac{7}{3} < x\).

Step 3: Solve Inequality on the Right

Now, solve the right part of the compound inequality \(3x - 1 < 8\). Add 1 to both sides to get \(3x < 9\). Then, divide both sides by 3 to isolate \(x\), yielding \(x < 3\).

Step 4: Combine Both Inequalities

We now have two inequalities: \(-\frac{7}{3} < x\) and \(x < 3\). Combine these to form a single compound inequality: \(-\frac{7}{3} < x < 3\).

Step 5: Write in Interval Notation

Express the solution set \(-\frac{7}{3} < x < 3\) in interval notation. Since \(x\) is greater than \(-\frac{7}{3}\) but less than 3, the interval is \((-\frac{7}{3}, 3)\).

Key Concepts

Interval Notation

Interval notation is a mathematical method used to describe the set of solutions to an inequality. It provides a concise representation of the beginning and end values of an interval. In our step-by-step exercise, we've used it to express the solution of the compound inequality.

Here's how it works:

• Round brackets like \( (a, b)\) indicate that both endpoints \(a\) and \(b\) are not included in the interval. This occurs when the inequality symbols are strict (greater than or less than).
• Square brackets like \[ [a, b] \] would indicate that the endpoints \(a\) and \(b\) are included, which happens with 'greater than or equal to' or 'less than or equal to'.
• The union of intervals can also be described using interval notation, where you might see the symbol \cup\ to denote union.

In our example, the solution is \( (-\frac{7}{3}, 3) \). The solution states that \(x\) lies strictly between \(-\frac{7}{3}\) and \(3\). Both \( -\frac{7}{3}\) and \(3\) are not included, as represented by rounded brackets. This is a perfect use of interval notation to express a continuous range of solutions.

Compound Inequalities

Compound inequalities involve combining two or more inequalities joined by 'and' or 'or'. They describe the range of values that satisfy all given conditions at once.

• If the inequalities are connected by 'and', like in our example, the solution is the overlap of the values that satisfy both inequalities. This is often seen as a segment on the number line.
• For 'or' inequalities, the solution includes any value that satisfies either inequality. This usually covers two or more separate intervals.

In the original step-by-step solution, we had \(-8 < 3x - 1 < 8\), which was split into two inequalities: \(-8 < 3x - 1\) and \(3x - 1 < 8\). By solving these two separately and combining them, we get the compound inequality \(-\frac{7}{3} < x < 3\). The solution is a single interval where both conditions are true. Compound inequalities help in identifying these solution ranges precisely, ensuring clarity in algebraic problem-solving.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations like addition, subtraction, multiplication, and division. They form the backbone of solving equations and inequalities, as seen in the given exercise.

Let's break down what's inside that absolute value:

• The expression \(3x - 1\) is what we refer to as an algebraic expression. It includes a variable \(x\) and constants combined through operations.
• To solve for \(x\), these expressions are manipulated using algebraic rules, such as add, subtract, divide, and multiply.

In the provided solution, we first rewrote the inequality without the absolute value bars. This step required understanding that \(|a| < b\) means that \(-b < a < b\). Then, the algebraic expressions on the left and right inequalities were simplified to isolate the variable \(x\). This essentially involved adding, subtracting, and dividing through algebraic manipulation. Mastering these basics enables students to tackle more complex problems with confidence.