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Chapter 1: Problem 46

Solve each inequality. Express your answer using set notation or interval notation. Graph the solution set. $$ |3 x+4| \geq 2 $$

Short Answer

Expert verified
The solution set is \(( -\infty, -2] \cup \left[ -\frac{2}{3}, \infty )\).

Step by step solution

01

- Understand the Absolute Value Inequality

The given inequality is \(|3x + 4| \geq 2\). Absolute value inequalities can be broken into two separate inequalities. If |A| \geq B, then A \geq B or A \leq -B.
02

- Set Up Two Inequalities

Write the inequality as two separate inequalities: 1. \(3x + 4 \geq 2\) 2. \(3x + 4 \leq -2\)
03

- Solve the First Inequality

Solve \(3x + 4 \geq 2\):Subtract 4 from both sides:\(3x \geq -2\).Divide by 3:\(x \geq -\frac{2}{3}\).
04

- Solve the Second Inequality

Solve \(3x + 4 \leq -2\):Subtract 4 from both sides:\(3x \leq -6\).Divide by 3:\(x \leq -2\).
05

- Combine the Solutions

Combine the solutions from Steps 3 and 4. The solution set is \(( -\infty, -2] \cup \left[ -\frac{2}{3}, \infty )\).
06

- Graph the Solution Set

Draw a number line with filled circles at \(x = -2\) and \(x = -\frac{2}{3}\). Shade the region to the left of \(x = -2\) and the region to the right of \(x = -\frac{2}{3}\).

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Key Concepts

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

absolute value inequalities
An absolute value inequality, like \(|3x + 4| \geq 2\), involves determining the distance of a value from zero on a number line. The absolute value function ensures that the expression inside is always positive or zero. When solving these inequalities, split the inequality into two parts:
  • \(A \geq B \Rightarrow 3x + 4 \geq 2\)
  • \(A \leq -B \Rightarrow 3x + 4 \leq -2\)
After splitting, solve each inequality separately. This helps in finding the two ranges that satisfy the given inequality.
interval notation
Interval notation is a way to describe the set of solutions for an inequality. After solving the inequalities above, you'll get two ranges. For this example, the ranges are \([x \geq -\frac{2}{3}]\) and \( [x \leq -2] \). These ranges can be rewritten using interval notation as follows:
  • For \(x \geq -\frac{2}{3}\): \([-\frac{2}{3}, \infty)\)
  • For \(x \leq -2: \): \((-\infty, -2]\)
Combined, they form the solution set in interval notation: \((-\infty, -2] \cup [-\frac{2}{3}, \infty)\). This notation effectively communicates the full range of possible solutions.
solution set
The solution set is the collection of all values that satisfy the inequality. For the inequality \(|3x + 4| \geq 2\), the steps involved splitting it into two inequalities and solving each, giving us two ranges: \([-\frac{2}{3}, \infty)\) and \((-\infty, -2]\)
  • Each range includes all numbers between its endpoints, and the endpoints themselves if the inequality is \(\backslashgeq\) or \(\backslashleq\).
  • Any real number in these intervals will satisfy the original inequality.
  • The combined intervals thus represent the complete solution set.
This explains why the combined intervals \((-\infty, -2] \cup [-\frac{2}{3}, \infty)\) form the solution set for our inequality.
graphing inequalities
Graphing inequalities helps visualize the solution set. For \(|3x + 4| \geq 2\), the solution set \((-\infty, -2] \cup [-\frac{2}{3}, \infty)\) is graphed on a number line. Here's how to do it:
  • Draw a horizontal line to represent the number line.
  • Mark the critical points \(-2\) and \(-\frac{2}{3}\).
  • Use filled circles at these points because the inequality includes the endpoints.
  • Shade to the left of \(-2\) to represent \(-\infty \leq x \leq -2\).
  • Shade to the right of \(-\frac{2}{3}\) for \(-\frac{2}{3} \leq x \leq \infty\).
The shaded regions show all possible solutions that satisfy the inequality visually. This makes it easier to understand the range of solutions.

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