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Chapter 4: Problem 54

Solve and graph each solution set. Write the answer using both set-builder notation and interval notation. $$ -6 \leq t+1 \text { and } t+8<2 $$

Short Answer

Expert verified
Set-builder: \{ t \mid -7 \leq t < -6 \}Interval notation: [-7, -6)

Step by step solution

01

Solve the first inequality

Start by solving the inequality \[\begin{equation} -6 \leq t+1 \end{equation}\] Subtract 1 from both sides: \[\begin{equation} -6 - 1 \leq t+1-1 \end{equation}\] This simplifies to: \[\begin{equation} -7 \leq t \end{equation}\]
02

Solve the second inequality

Solve the inequality \[\begin{equation} t+8 <2 \end{equation}\] Subtract 8 from both sides: \[\begin{equation} t+8-8 < 2-8 \end{equation}\] This simplifies to: \[\begin{equation} t < -6 \end{equation}\]
03

Find the intersection of the solutions

Combine the results of both inequalities: \[\begin{equation} -7 \leq t \end{equation}\] and \[\begin{equation} t < -6 \end{equation}\] The intersection is: \[\begin{equation} -7 \leq t < -6 \end{equation}\]
04

Write the answer in set-builder notation

The answer in set-builder notation is: \[\begin{equation} \{ t \mid -7 \leq t < -6 \} \end{equation}\]
05

Write the answer in interval notation

The answer in interval notation is: \[\begin{equation} [-7, -6) \end{equation}\]
06

Graph the solution set

On a number line, draw a closed circle at -7 and an open circle at -6. Shade the region between -7 and -6.

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Key Concepts

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

set-builder notation

Set-builder notation is a mathematical way of describing a set with a specific property. In this exercise, we want to solve the inequalities and express the solution set using this notation.
Here's the process:
  • Firstly, solve each inequality separately.
  • For \(-6 \leq t+1\), subtract 1 from both sides to get \(-7 \leq t\).
  • For \(t+8<2\), subtract 8 from both sides to get \(t<-6\).
  • The intersection of these solutions is \(-7 \leq t < -6\).
To express this using set-builder notation:
  • We use curly braces \(\{ \} \).
  • Within the braces, write the variable \(t\) followed by a vertical bar \( \mid \).
  • After the bar, write the condition \(-7 \leq t < -6\).
This gives us:
\( \{ t \mid -7 \leq t < -6 \} \)
This notation tells us that \(t\) is a set of numbers between \(-7\) and \(-6\), including \(-7\) but not \(-6\).
interval notation

Interval notation is another way to represent the set of solutions. It's often considered more compact and less verbose than set-builder notation.
Here's what you need to know:
  • Use brackets \([ \] and \(( \)) to denote the endpoints of the interval.
  • The bracket \([ \) means the endpoint is included, while the parenthesis \(( \)) means the endpoint is excluded.
  • In our solution \(-7\leq t<-6\), \(-7\) is included, and \(-6\) is not.
Thus, the interval notation for the solution is:
\(\[-7, -6\)\)
This compact form makes it easy to see the range of values for \(t\) at a glance.
In the context of this problem, both set-builder and interval notations are useful. They're just two different methods of representing the same solution set. Use whichever you're more comfortable with. Just ensure you're clear on the requirements of your assignment.
graphing solutions

Graphing solutions helps visualize the solution set on a number line. It's an excellent way to see the range of values that satisfy the inequality.
Follow these steps to graph the solution:
  • Draw a number line and mark the points \(-7\) and \(-6\).
  • At \(-7\), draw a closed circle because \(-7\) is included in the solution set.
  • At \(-6\), draw an open circle because \(-6\) is not included.
Once the points are marked, shade the region between \(-7\) and \(-6\).
This shading indicates every number in this interval is part of the solution. It's a straightforward way to show the values that make the inequality true.
Graphing is especially helpful for visual learners and can complement the set-builder and interval notations.
Keep practicing with different inequalities to make graphing solutions second nature.

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