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Magazine Chapter 2: Problem 121
Solve each inequality. Graph the solution set, and write it using interval notation. \(5<1-6 m<12\)
Short Answer
Expert verified
The solution is \((-\frac{11}{6}, -\frac{2}{3})\).
Step by step solution
01
Break Down the Compound Inequality
The given inequality is a compound inequality: \(5 < 1 - 6m < 12\)This can be broken down into two separate inequalities:1. \(5 < 1 - 6m\)2. \(1 - 6m < 12\)
02
Solve the First Inequality
First, solve \(5 < 1 - 6m\).- Subtract 1 from both sides: \(5 - 1 < -6m\)- Simplify: \(4 < -6m\)- Divide by -6 and remember to flip the inequality sign: \(\frac{4}{-6} > m\)- Simplify: \(-\frac{2}{3} > m\) or equivalently, \(m < -\frac{2}{3}\)
03
Solve the Second Inequality
Next, solve \(1 - 6m < 12\).- Subtract 1 from both sides: \(-6m < 12 - 1\)- Simplify: \(-6m < 11\)- Divide by -6 and remember to flip the inequality sign: \(m > \frac{11}{-6}\)Since \(\frac{11}{-6}\) simplifies to \(-\frac{11}{6}\), the inequality becomes \(m > -\frac{11}{6}\)
04
Combine the Inequalities
Combine the two results to get the final compound inequality:\(-\frac{11}{6} < m < -\frac{2}{3}\)
05
Graph the Solution Set
Graph the solution on a number line:- Draw a number line from \(-2\) to 0.- Use an open circle at \(-\frac{11}{6}\) (approximately -1.833) to indicate \(m > -\frac{11}{6}\).- Use an open circle at \(-\frac{2}{3}\) (approximately -0.667) to indicate \(m < -\frac{2}{3}\).- Shade the region between \(-\frac{11}{6}\) and \(-\frac{2}{3}\).
06
Write the Solution in Interval Notation
Express the solution in interval notation: \((-\frac{11}{6}, -\frac{2}{3})\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
interval notation
When solving inequalities, interval notation is a very useful way of representing a range of solutions. It uses parenthesis and brackets to show the start and end of the interval.
For our compound inequality \( -\frac{11}{6} < m < -\frac{2}{3} \), we use parenthesis because the inequality does not include the endpoints. Hence, the interval notation would be \( (-\frac{11}{6}, -\frac{2}{3}) \).
Remember:
For our compound inequality \( -\frac{11}{6} < m < -\frac{2}{3} \), we use parenthesis because the inequality does not include the endpoints. Hence, the interval notation would be \( (-\frac{11}{6}, -\frac{2}{3}) \).
Remember:
- Use parenthesis ( ) when the endpoint is not included, indicated by < or >.
- Use brackets [ ] when the endpoint is included, indicated by 鈮� or 鈮�.
graphing inequalities
Graphing inequalities helps visualize the range of values that satisfy the inequality. On a number line, open circles represent values that are not included in the solution, while closed circles represent values that are included.
For the inequality \( -\frac{11}{6} < m < -\frac{2}{3} \), you would draw:
For the inequality \( -\frac{11}{6} < m < -\frac{2}{3} \), you would draw:
- A number line extending from -2 to 0 (or beyond).
- Open circles at \( -\frac{11}{6} \) and \( -\frac{2}{3} \) to show these points are not included.
- A shaded region between these points indicating all the values that m can take.
algebraic manipulation
Understanding how to manipulate algebraic inequalities is essential for solving them. Here鈥檚 a quick recap on the steps:
1. **Breaking down compound inequalities**: Separate them into individual inequalities. For \( 5 < 1 - 6m < 12 \), break it down into \( 5 < 1 - 6m \) and \( 1 - 6m < 12 \).
2. **Solving each inequality**: Handle each part separately. For example:
By following these steps, you can confidently solve and express both simple and compound inequalities.
1. **Breaking down compound inequalities**: Separate them into individual inequalities. For \( 5 < 1 - 6m < 12 \), break it down into \( 5 < 1 - 6m \) and \( 1 - 6m < 12 \).
2. **Solving each inequality**: Handle each part separately. For example:
- Solve \( 5 < 1 - 6m \) by isolating m:
\begin{align*} 5 - 1 &< -6m \ 4 &< -6m \ \frac{4}{-6} &> m \ -\frac{2}{3} &> m \text{ (flip the inequality when dividing by a negative number)} \end{align*}This results in \( m < -\frac{2}{3} \). - Solve \( 1 - 6m < 12 \):
\begin{align*} -6m &< 12 - 1 \ -6m &< 11 \ m &> -\frac{11}{6} \text{ (again, flip the inequality when dividing by a negative number)} \end{align*}This results in \( m > -\frac{11}{6} \).
By following these steps, you can confidently solve and express both simple and compound inequalities.
