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Chapter 1: Problem 6

Write the given interval in set-builder notation. $$\left[-3, \frac{4}{3}\right)$$

Short Answer

Expert verified
\{ x | -3 \leq x < \frac{4}{3} \}

Step by step solution

01

- Identify the Interval

The given interval is \(\begin{bmatrix}-3, \frac{4}{3})\)}. It includes -3 and all the numbers up to but not including \(\frac{4}{3}\).
02

- Understand Set-Builder Notation

Set-builder notation is a way to describe a set by stating the properties that its members must satisfy. For intervals, it expresses the set of numbers (elements) that fall between endpoints with potential additional constraints.
03

- Translate to Set-Builder Notation

In set-builder notation, the interval \(\begin{bmatrix}-3, \frac{4}{3})\)} can be written as \(\begin{cases} \left\{ x \ | \ -3 \ \leq x \ < \frac{4}{3} \right\} \end{cases}\). This translates to the set of all x such that x is greater than or equal to -3 and less than \(\frac{4}{3}\).

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Key Concepts

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

Interval Notation
Intervals are used to describe a range of numbers between two endpoints. For example, the interval \([-3, \frac{4}{3})\) means that we include -3 up to but not including \(\frac{4}{3}\). This interval can be called 'half-open' because it includes one endpoint while excluding the other.
In writing interval notation:
  • The square bracket \[ \] means including the endpoint.
  • The parenthesis \( \) means excluding the endpoint.
Therefore, \([-3, \frac{4}{3})\) reads as all numbers starting from -3 and less than (but not equal to) \(\frac{4}{3}\).
Number Line
A number line visually represents sets of numbers along a single straight line. Each point on this line corresponds to a number. Visualizing intervals on a number line can help to understand their scope and extent.
For our example, the interval \([-3, \frac{4}{3})\) looks like this on a number line:
  • Draw a solid circle at -3 to indicate that -3 is included in the interval.
  • Draw an open circle at \(\frac{4}{3}\) to indicate that \(\frac{4}{3}\) is not included.
  • Shade the line segment between these two points to show all values in between.
By visualizing, we observe all real numbers from -3 up to (but not including) \(\frac{4}{3}\).
Inequalities
Inequalities are mathematical expressions that show the relationship between quantities that are not strictly equal. When we talk about intervals, we often use inequalities to describe them.
Inequalities include symbols like:
  • \(<\) for 'less than'
  • \(>\) for 'greater than'
  • \(\leq\) for 'less than or equal to'
  • \(\geq\) for 'greater than or equal to.'
For instance, the interval \([-3, \frac{4}{3})\) can be written as the inequality \(-3 \leq x < \frac{4}{3}\). This shows that \(x\) is between -3 and \( \frac{4}{3}\), including -3 but not including \( \frac{4}{3}.\)

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Most popular questions from this chapter

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The value of \(x\) differs from 4 by more than 1 unit.

Martha is in a boat in the ocean \(48 \mathrm{mi}\) from point \(A\), the closest point along a straight shoreline. She needs to dock the boat at a marina \(x\) miles farther up the coast, and then drive along the coast to point \(B, 96 \mathrm{mi}\) from point \(A\). Her boat travels \(20 \mathrm{mph}\), and she drives \(60 \mathrm{mph}\). If the total trip took \(4 \mathrm{hr}\), determine the distance \(x\) along the shoreline.

Use the relationship between temperature in Celsius and temperature in Fahrenheit. \(C=\frac{5}{9}(F-32)\) Body temperature is usually maintained between \(36.5^{\circ} \mathrm{C}\) and \(37.5^{\circ} \mathrm{C}\), inclusive. Determine the corresponding range of temperature in Fahrenheit.

Write the set as a single interval. $$[(-\infty,-2) \cup(4, \infty)] \cap[-5,3)$$

Write the solution set. a. \(|c-3|=0\) b. \(|c-3|<0\) c. \(|c-3| \leq 0\) d. \(|c-3|>0\) e. \(|c-3| \geq 0\)
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