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Chapter 8: Problem 68

Solve each inequality. Graph the solution set and write it using interval notation. \(0 \leq|6 x-3|-5\)

Short Answer

Expert verified
The solution in interval notation is \((-\infty, -\frac{1}{3}] \cup [\frac{4}{3}, \infty)\).

Step by step solution

01

Isolate the Absolute Value Expression

Start with the inequality \( 0 \leq |6x - 3| - 5 \). The first step is to isolate the absolute value expression on one side. To do this, add 5 to both sides of the inequality to remove the -5.\[0 + 5 \leq |6x - 3|\]This simplifies to:\[5 \leq |6x - 3|\]
02

Understand the Absolute Value Inequality

The inequality \( 5 \leq |6x - 3| \) states that the distance of \( 6x - 3 \) from 0 is at least 5. Absolute value inequalities of the form \( |A| \geq B \) can be split into two inequalities: \( A \geq B \) or \( A \leq -B \).
03

Split into Two Inequalities

Split \( |6x - 3| \geq 5 \) into:1. \( 6x - 3 \geq 5 \)2. \( 6x - 3 \leq -5 \)
04

Solve Each Inequality

Let's solve both inequalities for \( x \):**1. Solve** \( 6x - 3 \geq 5 \):- Add 3 to both sides:\[ 6x \geq 8 \]- Divide both sides by 6:\[ x \geq \frac{4}{3} \]**2. Solve** \( 6x - 3 \leq -5 \):- Add 3 to both sides:\[ 6x \leq -2 \]- Divide both sides by 6:\[ x \leq -\frac{1}{3} \]
05

Combine the Solutions

The solution to the inequality \( |6x - 3| \geq 5 \) is the combination of the solutions to both inequalities in Step 4. Thus, \( x \geq \frac{4}{3} \) or \( x \leq -\frac{1}{3} \).
06

Express in Interval Notation

Write the solution set in interval notation. Since there are two separate parts to the solution, the interval notation is:\[ (-\infty, -\frac{1}{3}] \cup [\frac{4}{3}, \infty) \]
07

Graph the Solution Set

To graph the solution set, draw a number line. Shade the region to the left of \(-\frac{1}{3}\), including \(-\frac{1}{3}\), using a closed dot. Then shade the region to the right of \(\frac{4}{3}\), including \(\frac{4}{3}\), also using a closed dot. This represents the values of \(x\) that satisfy the inequality.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation Explained
Imagine interval notation as a simple way to describe a set of numbers that form a solution to an inequality. It's like writing a shortcut for the numbers that work. Let's break down the interval notation:
  • Around brackets: "[ ]" means 'inclusive', showing the number is part of the solution.
  • Round brackets: "( )" means 'exclusive', meaning the number itself isn't part of the solution.
  • Infinity symbol (鈭�): Used to express that the values continue indefinitely in one direction.
For the inequality solution "\((-\infty, -\frac{1}{3}] \cup [\frac{4}{3}, \infty)\)", here's what it means:
  • "\((-\infty, -\frac{1}{3}]\)" represents all numbers less than or equal to \(-\frac{1}{3}\).
  • "\([\frac{4}{3}, \infty)\)" represents all numbersgreater than or equal to \(\frac{4}{3}\).
  • The union symbol "\(\cup\)" is like saying "or"; either condition satisfies the inequality.
Understanding interval notation is key to expressing solutions clearly and efficiently.
Graphing Inequalities: Visualizing Solutions
Graphing inequalities helps you visually interpret the solution on a number line. This is a powerful tool because it shows all possible numbers that solve the inequality. For the inequality determined in the original exercise, here's how you graph it:First, identify the critical points, which are \(-\frac{1}{3}\) and \(\frac{4}{3}\).
  • Place a closed dot on \(-\frac{1}{3}\) because the inequality includes this point ("\([\)]").
  • Shade the entire line to the left of this point, showing values less than or equal to \(-\frac{1}{3}\).
Next, place another closed dot on \(\frac{4}{3}\).
  • Again, shade the entire line to the right of this point, expressing values greater than or equal to \(\frac{4}{3}\).
Use these visual tools to ensure the interval notation and graph align, confirming accuracy in solution representation.
Solving Absolute Value Inequalities
Solving absolute value inequalities might seem tricky at first, but there's a straightforward approach to tackle them effectively. Absolute value refers to the distance a number is from zero, so it鈥檚 always positive or zero. Here's how to handle inequalities with absolute values, such as \(|6x - 3| \geq 5\):
  • Start by isolating the absolute value expression. Here, it鈥檚 already isolated as \(5 \leq |6x - 3|\).
  • This inequality \(|A| \geq B\) can be split into two separate inequalities: \(A \geq B\) and \(A \leq -B\).
For example:
  • Convert \(|6x - 3| \geq 5\) into two inequalities:
    • \(6x - 3 \geq 5\)
    • \(6x - 3 \leq -5\)
Solve each inequality:
  • For \(6x - 3 \geq 5\), solve \(x \geq \frac{4}{3}\).
  • For \(6x - 3 \leq -5\), solve \(x \leq -\frac{1}{3}\).
Joining these solutions gives the complete set of solutions \(x \geq \frac{4}{3}\) or \(x \leq -\frac{1}{3}\).By understanding these steps, you can confidently solve absolute value inequalities and express the solutions using interval notation or on a number graph.

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