Problem 36 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Give the solution set using interval notation. $$|7-3 x|>4$$

Step-by-Step Solution

Answer

The solution set is \( x \in (-\infty, 1 ) \cup (\frac{11}{3}, \infty) \).

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|7-3x| > 4 \). Absolute value inequalities \(|A| > B \) can be split into two inequalities: \(A > B \) and \(A < -B \).

Step 2: Split into Two Inequalities

Split \( |7 - 3x| > 4 \) into two separate inequalities: \(7 - 3x > 4 \) and \(7 - 3x < -4 \).

Step 3: Solve the First Inequality

Solve \( 7 - 3x > 4 \): \(-3x > 4 - 7 \), \(-3x > -3 \), \(x < 1 \) (remember to reverse inequality sign when dividing by a negative).

Step 4: Solve the Second Inequality

Solve \( 7 - 3x < -4 \): \(-3x < -4 - 7 \), \(-3x < -11 \), \(x > \frac{11}{3} \) (again, reverse inequality sign when dividing by a negative).

Step 5: Combine the Solutions

Combine the inequalities from Steps 3 and 4. The solution set for \( |7 - 3x| > 4 \) is: \( x \in (-\infty, 1 ) \cup (\frac{11}{3}, \infty) \).

Key Concepts

inequality solving

In order to solve inequalities, we start by isolating the variable on one side of the inequality. Remember, whenever you multiply or divide by a negative number, you must reverse the direction of the inequality sign.
Here is a closer look at the process:
1. **Isolate the Variable**: Move the variable to one side, and any constants to the other.
2. **Manipulate the Inequality**: Perform the same operations on both sides of the inequality, such as adding, subtracting, or dividing by the same number.
3. **Reverse the Sign When Necessary**: Whenever you multiply or divide by a negative number, flip the inequality sign.
Let's apply these steps to our exercise:
For the inequality \(7 - 3x > 4\), subtract 7 from both sides, and divide by -3, remembering to reverse the inequality sign, giving you \(x < 1\).
For \(7 - 3x < -4\), similarly subtract 7 from both sides, divide by -3, and reverse the inequality, giving you \((x > \frac{11}{3})\). Finally, combine these results for the full solution. This systematic approach helps in efficiently solving any inequality you encounter.

interval notation

Interval notation is a way of writing subsets of the real number line. It is easy to understand and concise.
Here's how we express ranges using interval notation:

• **Open Interval** \( (a, b) \): Includes all the numbers between a and b, but not a and b.
• **Closed Interval** \( [a, b] \): Includes all numbers between a and b, including a and b.
• **Half-Open Interval** \( (a, b] \) or \( [a, b) \): Includes all numbers between a and b, but one of the endpoints is not included.
• **Unbounded Intervals**: Use infinity \( \infty \) or negative infinity \( -\infty \) to indicate that there is no bound in one direction. For example, \( (-\infty, a) \) and \( (b, \infty] \).

In the given exercise, the solutions are in open intervals because the endpoints are not included. Thus, combining \( x < 1 \) and \( x > \frac{11}{3}\), we write: \( x \in (-\infty, 1) \cup (\frac{11}{3}, \infty) \). This notation is compact and effectively communicates the solution set.

absolute value properties

Absolute value inequalities are special cases that deal with the distance from zero on a number line. The absolute value of a number \( |x| \) measures how far x is from 0, regardless of direction.
An absolute value inequality will generally split into two separate inequalities:

• \{|A| > B \} becomes \{A > B \} or \{A < -B \} because either side of the absolute value can be greater than B.
• \{|A| < B \} becomes \{-B < A < B \} because the value is trapped within the bounds of B in both positive and negative directions.

For our exercise, \{|7-3x| > 4 \}, we split into \{7-3x > 4 \} and \{7-3x < -4 \}. Solving each inequality will then give us ranges that we ultimately combine using union notation (\cup). Understanding the properties of absolute values helps break any complex absolute value inequality into simpler parts, making it easier to solve.