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Chapter 1: Problem 36

Solve and write interval notation for the solution set. Then graph the solution set. $$2 x<8 \text { or } x+3 \geq 10$$

Short Answer

Expert verified
The solution set in interval notation is \( (-\infty, 4) \cup [7, \infty)\).

Step by step solution

01

- Solve the First Inequality

Solve the inequality \(2x < 8\). Divide both sides by 2: \( x < 4\). This is the solution to the first inequality.
02

- Solve the Second Inequality

Solve the inequality \(x + 3 \geq 10\). Subtract 3 from both sides: \(x \geq 7\). This is the solution to the second inequality.
03

- Combine the Solutions

Combine the solution sets for the two inequalities using the 'or' condition. The solutions are \(x < 4\) and \(x \geq 7\). This means we have two intervals: \( (-\infty, 4) \cup [7, \infty)\)
04

- Graph the Solution Set

To graph the solution set, draw a number line. Shade the region to the left of 4 (not including 4) and shade the region to the right of 7 (including 7).

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Key Concepts

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

Interval Notation
Interval notation provides a way to describe sets of numbers between given boundaries. It uses parentheses \( () \) for values that are not included and brackets \([聽]\) for values that are included.
In the given example, we have two inequalities: \( 2x < 8 \) and \( x + 3 \geq 10 \). Solving them, we get \( x < 4 \) and \( x \geq 7 \.\)
In interval notation:
- \( x < 4 \) includes all numbers less than 4, so we write \( ( -\infty, 4) \).
- \( x \geq 7 \) includes 7 and all numbers greater than 7, so we write \( [7, \infty) \.\)
The solutions are combined using the 'or' condition, giving us the interval notation: \(( -\infty , 4) \cup [7, \infty) \).
Graphing Inequalities
Graphing inequalities involves representing the solution set on a number line.
To do this, follow these steps:
  • Draw a number line.
  • Identify the points of interest. Place points at 4 and 7 in our example.
  • Shade the region to the left of 4 to represent all numbers less than 4. Use an open circle at 4 because it is not included (as per \( < 4 \)).
  • Shade the region to the right of 7 to represent all numbers greater than or equal to 7. Use a closed circle at 7 because it is included (as per \( \geq 7 \)).
Combining these, you get a graph with shaded regions on both ends of the number line: from \( -\infty \) to 4 (not including 4) on the left side and from 7 (including 7) to \( \infty \) on the right side.
Compound Inequalities
Compound inequalities are combinations of two or more inequalities joined by 'and' or 'or'.
In our example, we have \( 2x < 8 \) or \( x + 3 \geq 10 \). Solving these:
- For \( 2x < 8 \), we divide both sides by 2, giving \( x < 4 \).
- For \( x + 3 \geq 10 \), we subtract 3 from both sides, giving \( x \geq 7 \.\)
Since the inequalities are connected by 'or', the solution set includes numbers that satisfy either inequality. Thus, any number less than 4 or greater than or equal to 7 works.
In summary, compound inequalities help us find solutions that may span across different intervals, simplifying the process of solving multiple inequalities at once.

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