Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation or inequality. $$|3 x-1|<8$$

Step-by-Step Solution

Answer

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

Step 1: Understanding Absolute Inequality

This problem asks us to solve a strict inequality involving an absolute value. The expression to solve is \(|3x - 1| < 8\). The absolute inequality \(|A| < B\) can be rewritten as two separate inequalities: \[ -B < A < B \].

Step 2: Split the Inequality

Replace \(|3x - 1| < 8\) to its equivalent double inequality:\[-8 < 3x - 1 < 8\]. This gives us two conditions that must be satisfied simultaneously.

Step 3: Solving the Left Inequality

Start with the inequality: \[-8 < 3x - 1\]. Add 1 to both sides to isolate terms with x: \[-8 + 1 < 3x\], which simplifies to \[-7 < 3x\]. Dividing both sides by 3 to solve for x, we get: \[-\frac{7}{3} < x\].

Step 4: Solving the Right Inequality

Now consider the second inequality: \[3x - 1 < 8\]. Add 1 to both sides to isolate terms with x: \[3x < 9\]. Divide both sides by 3 to find x: \[x < 3\].

Step 5: Combine the Inequalities

From both inequalities, we have the compound solution: \[-\frac{7}{3} < x < 3\]. This means that x is greater than \(-\frac{7}{3}\) and less than 3.

Key Concepts

Absolute Value

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It's denoted by two vertical bars, for example, \( |x| \). This means that both \( |5| \) and \( |-5| \) equal 5. When dealing with absolute values in equations or inequalities, we're essentially considering two scenarios: one where the value inside the absolute bars is positive, and one where it's negative.

To solve an absolute value inequality like \( |3x - 1| < 8 \), we consider both the positive and negative aspects of the expression inside the bars. This gives us two simultaneous inequalities to solve: \(-8 < 3x - 1 < 8\). Understanding this concept is key, as it helps us express the solution as a range within which x can vary.

Compound Inequalities

Compound inequalities contain two separate inequalities joined by the word "and" or "or." In the case of absolute value inequalities that utilize a less than sign, such as \( |3x-1| < 8 \), they become compound inequalities that imply an "and" relationship.

For example, from \[|3x - 1| < 8\], we get the compound inequality: \(-8 < 3x - 1 < 8\). This "and" relationship signifies both conditions must be true at the same time.

To solve it, we handle each part of the inequality separately but are mindful that both conditions need to be satisfied by the same set of x values.

• The first part: \(-8 < 3x - 1\).
• The second part: \(3x - 1 < 8\).

Both of these inequalities define the set of real numbers x can be, creating a segment or interval on the number line where the solution lies.

Inequality Solving

Solving inequalities is similar to solving equations but with extra care towards the direction of the inequality. Just like how we solve equations, we apply operations to both sides to isolate the variable. However, we must remember:

• Adding or subtracting a number won't change the inequality direction.
• Multiplying or dividing by a positive number also keeps the direction unchanged.
• Multiplying or dividing by a negative number will flip the inequality sign.

In our exercise, we start with the two inequalities derived from the absolute value inequality:

• For \(-8 < 3x - 1\), adding 1 gives \(-7 < 3x\). Dividing by 3, x becomes \(-\frac{7}{3} < x\).
• For \(3x - 1 < 8\), adding 1 yields \(3x < 9\). Dividing by 3 gives us \(x < 3\).

The intersection of these two solutions calls for x to satisfy both conditions \(-\frac{7}{3} < x < 3\). This process exemplifies how to systematically solve inequalities while highlighting the importance of understanding inequality properties.