Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 55
Solve each inequality. Graph the solution set and write it in interval notation. $$ -3<3 x<6 $$
Short Answer
Expert verified
The solution set is \((-1, 2)\).
Step by step solution
01
Understand the Compound Inequality
The inequality given is \(-3 < 3x < 6\). This means we need to find all the values of \(x\) that satisfy both \(-3 < 3x\) and \(3x < 6\) simultaneously.
02
Solve the Left Inequality
First, solve \(-3 < 3x\) by isolating \(x\). Divide both sides by 3 to get \(-1 < x\).
03
Solve the Right Inequality
Next, solve \(3x < 6\) by isolating \(x\). Divide both sides by 3 to get \(x < 2\).
04
Combine the Solutions
The solutions of the compound inequality are \(-1 < x\) and \(x < 2\). Combining these, the solution is \(-1 < x < 2\).
05
Write in Interval Notation
The interval notation for \(-1 < x < 2\) is \((-1, 2)\).
06
Graph the Solution
To graph the solution \((-1, 2)\), draw a number line. Place open circles (indicating that the endpoint is not included) at \(-1\) and \(2\), and shade the line between these two points.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Compound Inequalities
Compound inequalities bring together two inequalities into a single statement, often using the word "and". The goal is to find the values that satisfy both inequalities simultaneously. For the compound inequality \-3 < 3x < 6\, you need to address each part separately before combining them into a complete solution.
- First, solve the left side: \-3 < 3x\. You want to isolate the variable \( x \), so divide both sides by 3. This gives you \-1 < x\.
- Next, solve the right side: \(3x < 6\). Likewise, divide by 3 to isolate \( x \), providing \( x < 2 \).
- Finally, put the two parts together: \-1 < x < 2\.
Interval Notation Explained
Interval notation is a concise way of expressing a range of numbers, making it a favorite in mathematics. When you express a solution in interval notation, you are detailing exactly what numbers are included in the solution set.
For the compound inequality \-1 < x < 2\, the interval notation is written as \((-1, 2)\). Here's the breakdown:
For the compound inequality \-1 < x < 2\, the interval notation is written as \((-1, 2)\). Here's the breakdown:
- The round brackets \(( )\) mean that the endpoints \(-1\) and \(2\) are not included in the solution (this is known as "not inclusive"). If the endpoints were included, square brackets \([ ]\) would be used instead.
- The order reflects the range from left to right on a number line, starting with the smallest number \(-1\) and ending with the largest number \(2\).
Graphing Inequalities on a Number Line
Graphing involves visually representing the range of solutions on a number line, which helps in understanding the solution better.
For the interval \((-1, 2)\):
For the interval \((-1, 2)\):
- Draw a horizontal line to represent the number line.
- Mark the points \(-1\) and \(2\) on this line.
- Since the endpoints are not included, draw open circles around \(-1\) and \(2\). Open circles indicate that these points themselves are excluded from the solution set.
- Shade the region between \(-1\) and \(2\) to show all the numbers that are part of the solution. This visually highlights \(-1 \lt x \lt 2\).
Solving Inequalities Step by Step
To solve an inequality, it's essential to manipulate it in a way that isolates the variable, similar to solving an equation. However, the rules for inequalities include a special caveat about direction changes.
- When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips. Thankfully, in our example, this wasn't necessary, as we divided by a positive \(3\).
- Steps for solving involve isolating the variable \( x \) either by adding, subtracting, multiplying, or dividing both sides by the same number.
- Consider both parts of the compound inequality separately first, as we combined the solutions \(-1 < x\) and \(x < 2\) to form the final interval \(-1 < x < 2\).
