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Chapter 4: Problem 22

Graph each inequality, and write the solution set using both set-builder notation and interval notation. $$ x \geq-6 $$

Short Answer

Expert verified
\( x \, \geq \, -6 \) in set-builder notation is \( \{ x \, | \, x \, \geq \, -6 \} \) and in interval notation is \( [-6, \, \infty) \).

Step by step solution

01

Understanding the Inequality

The given inequality is \( x \, \geq \, -6 \). This means that x can be any number that is greater than or equal to -6.
02

Graphing the Inequality

To graph the inequality on a number line, draw a number line and place a closed circle at -6 to show that -6 is included in the solution. Then, shade the region to the right of -6 to represent all numbers greater than -6.
03

Writing the Solution in Set-Builder Notation

In set-builder notation, the solution set is written as: \[ \{ x \, | \, x \, \geq \, -6 \} \]
04

Writing the Solution in Interval Notation

In interval notation, the solution set is written as: \[ [-6, \, \infty) \]

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Key Concepts

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

inequality notation
Inequality notation is used to represent a range of values that satisfy an inequality. In the given exercise, you see the inequality: \(x \geq -6\). This tells us that \(x\) can be any number that is greater than or equal to -6. When using inequality notation:
  • The symbol \(<\) means 'less than.'
  • The symbol \(>\) means 'greater than.'
  • The symbol \(\leq\) means 'less than or equal to.'
  • The symbol \(\geq\) means 'greater than or equal to.'
For our example, \(x \geq -6\) includes every number from -6 and beyond to positive infinity.
number line representation
A number line is a visual way to represent solutions to inequalities. To graph \(x \geq -6\) on a number line:
  • Draw a horizontal line and mark numbers on it including -6.
  • Put a closed circle (or dot) at -6 because \(-6\) is included (\(x\) can equal -6).
  • Shade the region to the right of -6 to show that all numbers greater than -6 are also solutions.
The closed circle means -6 is part of the solution. The shaded area to the right represents all possible values that are greater than -6. This visualization helps to clearly see the range of possible values that satisfy the inequality.
set-builder notation
Set-builder notation is a concise way to describe a set of numbers that satisfy a condition. For our exercise, the set of numbers that satisfy the inequality \(x \geq -6\) can be written in set-builder notation as:
\[ \{ x \, | \, x \, \geq \, -6 \} \]
Here's how to read it:
  • The curly braces \(\{ \}\) denote a set.
  • The variable \(x\) represents the elements of the set.
  • The vertical bar \(|\) can be read as 'such that.'
  • The expression \(x \geq -6\) states the condition that elements of the set must satisfy.
So, \(\{ x \, | \, x \, \geq \, -6 \}\) means 'the set of all \(x\) such that \(x\) is greater than or equal to -6.'
interval notation
Interval notation is a shorthand way of describing a continuous range of values. The solution to the inequality \(x \geq -6\) can be written in interval notation as: \[ [-6, \, \infty) \]
In interval notation:
  • The bracket \([\) means the interval includes -6 (closed interval).
  • The comma separates the lower and upper bounds.
  • The infinity symbol (\(\infty\)) indicates that there is no upper limit.
  • The parenthesis \()\) means infinity is not included (open interval).
This shows that the solution includes all numbers from -6 up to and including infinity, but infinity itself is not a specific number and hence is not included.

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Most popular questions from this chapter

The function given by $$ P(d)=1+\frac{d}{33} $$ gives the pressure, in atmospheres \((a t m),\) at a depth of \(d\) feet in the sea. For what depths \(d\) is the pressure at least 1 atm and at most 7 atm?

Waterfalls. In order for a waterfall to be classified as a classical waterfall, its height must be less than twice its crest width, and its crest width cannot exceed one-and-one-half times its height. The tallest waterfall in the world is about 3200 ft high. Let \(h\) represent a waterfall's height, in feet, and w the crest width, in feet. Write and graph a system of inequalities that represents all possible combinations of heights and crest widths of classical waterfalls.

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=\sqrt{8-5 x} $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=\sqrt{2 x+7} $$

For \(f(x)\) as given, use interval notation to write the domain of \(f\). $$ f(x)=-\frac{6}{x} $$
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