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Chapter 1: Problem 9

Write the given inequality using interval notation and then graph the interval. $$ x \geq 5 $$

Short Answer

Expert verified
The interval notation is \([5, \, \infty)\). Graph: Solid dot on 5, shade right.

Step by step solution

01

Understand the inequality

The given inequality is \( x \geq 5 \). This means that \( x \) can be any number that is greater than or equal to 5.
02

Convert to Interval Notation

In interval notation, inequality \( x \geq 5 \) is written as \([5, \, \infty)\). This is because the interval includes 5 (hence the square bracket) and extends indefinitely towards positive infinity.
03

Graph the Interval

To graph \([5, \, \infty)\): Draw a number line, place a solid dot on 5 to indicate that 5 is included in the interval, and shade the line extending to the right towards positive infinity.

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Key Concepts

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

Graphing Inequalities
When dealing with inequalities, it's important to understand how to visualize them through graphing. Inequalities such as \( x \geq 5 \) describe a range of values rather than a single point. This specific inequality means that \( x \) can be any number greater than or equal to 5. Graphing this on a number line helps provide a clear picture:
  • First, draw a straight horizontal line to represent possible values of \( x \).
  • Then, locate the number 5 on this line, since it鈥檚 the point of interest.
  • To show \( x \geq 5 \), place a solid dot at 5. This indicates that the number 5 is included in the solution set (hence solid, not open).
  • Shade the line to the right of the dot. This shading shows that all numbers greater than 5 are also part of the solution set.
By graphing, we can easily communicate which numbers satisfy the inequality without having to list them all. The solid dot and the shaded region together provide a clear and visual solution.
Interval Notation
Interval notation is a mathematical notation system used to express a set of numbers. In the context of inequalities, it helps succinctly convey which numbers are included in the solution set.
  • For the inequality \( x \geq 5 \), the interval includes 5 and all numbers greater than 5.
  • When writing this in interval notation, we represent it as \([5, \, \infty)\).
  • The square bracket \([ \) signifies that 5 is included in the interval, distinguishing it from numbers less than 5 which are not part of the solution.
  • The round parenthesis \()\) following infinity \((\infty)\) means that infinity is not a number that can be reached or included.
Interval notation is compact and efficient, providing a quick and clear way to describe the range of numbers in solutions to inequalities.
Number Line Representation
Representing inequalities on a number line is one of the most intuitive ways to showcase solutions to inequalities. It gives a straightforward visual representation of all potential solutions, which can be very helpful when trying to understand or convey the concept.
  • A number line is a straight horizontal line used to visually represent numbers and intervals.
  • To depict the inequality \( x \geq 5 \):
  • Draw a number line with evenly spaced points marking integers, including the number 5.
  • Place a solid dot directly on the 5. This indicates that 5 is indeed a part of the solution set - it meets the criteria \( \geq \) (greater than or equal to).
  • Shade a line extending from the solid dot on 5 to the right, indicating that all numbers greater than 5 are also included.
Using a number line allows one to quickly see all viable solutions to an inequality. In this case, any number starting from 5 onwards qualifies, perfectly illustrated by the shaded region on the number line. The number line representation is particularly useful in educational settings for its simplicity and clarity.

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Most popular questions from this chapter

Use factorization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part \((b)\). (a) \(\frac{x^{2}+x-6}{x^{2}-5 x+6}\) (b) \(\lim _{x \rightarrow 2} \frac{x^{2}+x-6}{x^{2}-5 x+6}\)

Use binomial expansion to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{5-5(h+1)^{2}}{h}\) (b) \(\lim _{h \rightarrow 0} \frac{5-5(h+1)^{2}}{h}\)

Use rationalization to simplify the given expression in part (a). Then, if instructed, find the indicated limit in part (b). (a) \(\frac{4 y^{2}}{\sqrt{y^{2}+y+1}-\sqrt{y+1}}\) (b) \(\lim _{y \rightarrow 0} \frac{4 y^{2}}{\sqrt{y^{2}+y+1}-\sqrt{y+1}}\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y=|x-9|\)

Find any intercepts of the graph of the given equation. Determine whether the graph of the equation possesses symmetry with respect to the \(x\) -axis, \(y\) -axis, or origin. Do not graph. \(y^{3}-4 x^{2}+8=0\)
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