Problem 80 Solve and graph. Write the answe... [FREE SOLUTION]

Chapter 4: Problem 80

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ \left|\frac{2-5 x}{4}\right| \geq \frac{2}{3} $$

Short Answer

Expert verified

In set-builder notation: $ \{ x \ | \ x \leq -\frac{2}{15} \text{ or } x \geq \frac{14}{15} \} $. In interval notation: $ \left( -\infty, -\frac{2}{15} \right] \cup \left[ \frac{14}{15}, \infty \right) $.

Step by step solution

- Understand the Absolute Value Inequality

We are given the inequality \ \ $ \left|\frac{2-5x}{4}\right| \geq \frac{2}{3} $. An absolute value inequality of the form $ |A| \geq B $ splits into two inequalities: $ A \geq B $ or $ A \leq -B $.

- Set Up the Inequalities

Set up two separate inequalities from $ \left|\frac{2-5x}{4}\right| \geq \frac{2}{3} $: \ \ $ \frac{2-5x}{4} \geq \frac{2}{3} $ \ or \ $ \frac{2-5x}{4} \leq -\frac{2}{3} $.

- Solve the First Inequality

Solve $ \frac{2-5x}{4} \geq \frac{2}{3} $: \ \ Multiply both sides by 4: \ $ 2 - 5x \geq \frac{8}{3} $. \ \ Subtract 2 from both sides: \ $ -5x \geq \frac{8}{3} - 2 $. \ \ Find a common denominator for 2 (which is 3): \ $ -5x \geq \frac{8}{3} - \frac{6}{3} $. \ Simplify: \ $ -5x \geq \frac{2}{3} $. \ \ Divide by -5 and reverse the inequality: \ $ x \leq -\frac{2}{15} $.

- Solve the Second Inequality

Solve $ \frac{2-5x}{4} \leq -\frac{2}{3} $: \ \ Multiply both sides by 4: \ $ 2-5x \leq -\frac{8}{3} $. \ \ Subtract 2 from both sides: \ $ -5x \leq -\frac{8}{3} - 2 $. \ \ Find a common denominator for 2 (which is 3): \ $ -5x \leq -\frac{8}{3} - \frac{6}{3} $. \ Simplify: \ $ -5x \leq -\frac{14}{3} $. \ \ Divide by -5 and reverse the inequality: \ $ x \geq \frac{14}{15} $.

- Combine Solutions

Combine both parts to get the complete solution: \ $ x \leq -\frac{2}{15} $ or $ x \geq \frac{14}{15} $.

- Write in Set-Builder and Interval Notation

Set-builder notation: $ \{x \ | \ x \leq -\frac{2}{15} \text{ or } x \geq \frac{14}{15} \} $. \ Interval notation: $ \left( -\infty, -\frac{2}{15} \right] \cup \left[ \frac{14}{15}, \infty \right) $.

- Graph the Solution

Graph the intervals on a number line. Use a closed dot at $ -\frac{2}{15} $ and $ \frac{14}{15} $, and shade to the left of $ -\frac{2}{15} $ and to the right of $ \frac{14}{15} $ to represent $ \left( -\infty, -\frac{2}{15} \right] \cup \left[ \frac{14}{15}, \infty \right) $.

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Key Concepts

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

absolute value

Absolute value represents the distance a number is from zero on the number line, regardless of direction. For instance, $|3| = 3$ and $|-3| = 3$. It's always non-negative. In absolute value inequalities like $|A| \geq B$, the inequality splits into two: $A \geq B$ or $A \leq -B$. This is crucial for solving the inequality $\left|\frac{2-5x}{4}\right| \geq \frac{2}{3}$. Breaking it down this way ensures we capture all possible solutions.

interval notation

Interval notation is a way to describe sets of numbers. It uses brackets and parentheses to show where intervals start and end.

Square brackets [ ] mean the endpoint is included (closed interval).
Parentheses ( ) mean the endpoint is not included (open interval).
For example, $[1, 5)$ includes all numbers from 1 to 5, including 1 but not 5.
In our problem, the solution $ \left( -\infty, -\frac{2}{15} \right] \cup \left[ \frac{14}{15}, \infty \right) $ combines two intervals, showing where the solution lies on the number line.

Using interval notation helps us easily represent ranges of values. For the absolute value inequality, it makes the solution clear by indicating no gaps between intervals.

set-builder notation

Set-builder notation is another way to describe sets of numbers. It uses a rule to specify the properties of the elements in the set.

General form: $ \{ x \mid \text{condition} \} $.
The symbol $|$ (read as 'such that') defines the rule.

For example, $ \{ x \mid x \geq 2 \} $ means all $x$ such that $x$ is greater than or equal to 2. In our problem, the solution $ \{ x \mid x \leq -\frac{2}{15} \text{ or } x \geq \frac{14}{15} \} $ states all $x$ values that satisfy the inequality. This notation clearly specifies the range of solutions without listing all numbers.

graphing inequalities

Graphing inequalities helps visualize solutions on a number line.

Identify critical points from the solution. In our case, they are $ -\frac{2}{15} $ and $ \frac{14}{15} $.
Use closed dots to represent inclusive endpoints (e.g. $ \leq $ or $ \geq $).
Shade regions that satisfy the inequality. For $x \leq -\frac{2}{15}$, shade left of $ -\frac{2}{15} $. For $x \geq \frac{14}{15}$, shade right of $ \frac{14}{15} $.

Visual representation helps understand the solution set and how it spreads across the number line. This makes abstract concepts like inequalities more concrete and easier to grasp.

91影视

Solve and graph. Write the answer using both set-builder notation and interval notation. $$ \left|\frac{2-5 x}{4}\right| \geq \frac{2}{3} $$

Short Answer

Step by step solution

- Understand the Absolute Value Inequality

- Set Up the Inequalities

- Solve the First Inequality

- Solve the Second Inequality

- Combine Solutions

- Write in Set-Builder and Interval Notation

- Graph the Solution

Key Concepts

absolute value

interval notation

set-builder notation

graphing inequalities

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