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Chapter 2: Problem 109

Graph all solutions on a number line and provide the corresponding interval notation. $$ x<5 \text { or } x \geq 15 $$

Short Answer

Expert verified
The solution is \((- fty, 5) \cup [15, fty)\).

Step by step solution

01

Identify Inequalities

Look at the inequalities separately. The first inequality is \( x < 5 \) and the second is \( x \geq 15 \). These will guide us in determining which parts of the number line to highlight.
02

Graph \( x < 5 \) on the Number Line

For \( x < 5 \), draw a number line and create an open circle on 5 to indicate that 5 is not included. Shade everything to the left of 5 to indicate that all numbers less than 5 are part of the solution.
03

Graph \( x \geq 15 \) on the Number Line

For \( x \geq 15 \), draw a number line and create a closed circle at 15 to indicate inclusion. Shade everything to the right of 15, including 15.
04

Combine Graphs on the Number Line

Combine the shaded regions from \( x < 5 \) and \( x \geq 15 \) onto one number line. The parts of the number line less than 5 and greater than or equal to 15 are the solution areas.
05

Write Interval Notation

The interval notation for \( x < 5 \) is \((-fty, 5)\) and for \( x \geq 15 \) is \([15, fty)\). Combine these intervals to express the complete solution: \((-fty, 5) \cup [15, fty)\).

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Key Concepts

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

Interval Notation
Interval notation is a handy way to express the set of solutions for an inequality. It's all about visually capturing the start and end of a range.
For the inequality \(x < 5\), you use an open parenthesis \((-\infty, 5)\) because 5 is not included in the solution set. The open parenthesis signifies that the endpoint is not part of the interval.
For \(x \geq 15\), a closed bracket \([15, \infty)\) is used, as 15 is included in the solution.
  • Open parentheses \((\) are used when the number itself is not part of the solution.
  • Closed brackets \([\) signify that the number is included.
You often see \(\infty\) or \(-\infty\) in interval notation. These symbols represent unbounded intervals. It's important to remember that \(\infty\) and \(-\infty\) always have open parentheses, as they can never truly be reached.
Number Line
A number line is a simple yet powerful tool to visualize inequalities. It allows you to quickly understand which numbers lie within a certain range. For \(x < 5\), place an open circle at 5 and shade everything left of it. This shows none of the shaded numbers include 5, capturing the essence of "less than".
For \(x \geq 15\), use a closed circle at 15 and shade everything to the right, indicating 15 is part of the solution.
  • An open circle implies the number is not included (e.g., \(x < 5\)).
  • A closed circle implies inclusion (e.g., \(x \geq 15\)).
When these visual clues are combined on a single number line, it's easy to see the solution which might look scattered but represents all possible values satisfying both inequalities.
Graphing Inequalities
Graphing inequalities on a number line is a way of translating algebraic expressions into visual formats. This makes it easier to identify solution sets.For the inequality \(x < 5\), drawing it involves placing an open circle at 5, shading everything left to show all numbers less than 5 are included. For \(x \geq 15\), a closed circle at 15 with shading to the right shows inclusion of 15 and everything greater.
  • Graphs help in identifying overlapping or separate solutions.
  • They visually display the range of possible solutions for any given inequality.
Graphing multiple inequalities helps highlight the complete set of solutions and allows for easy interpretation of complex inequalities like \(x < 5 \text{ or } x \geq 15\). It's about making abstract concepts concrete and accessible.

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