Question
Solve the inequality, Graph the solution set, and write the solution set in set-buider notation and interval notation. Write numbers as simplified fractions or integers. \[ -3 m-4>23 \]
Ask by Little Daniel. in the United States
Feb 25,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The solution to the inequality \(-3m - 4 > 23\) is \(m < -9\). In set-builder notation, this is \(\{ m \in \mathbb{R} \mid m < -9 \}\), and in interval notation, it is \((-\infty, -9)\). On a number line, this is represented by an open circle at -9 and shading to the left.
Solution
Step 1. Solve the inequality
We start with the inequality:
-3m - 4 > 23
First, add 4 to both sides:
-3m - 4 + 4 > 23 + 4
-3m > 27
Next, divide both sides by -3. Remember, when dividing by a negative number, the inequality sign reverses:
m < 27/(-3)
m < -9
So the solution is: m < -9.
Step 2. Write the solution set in set-builder and interval notations
• Set-builder notation: { m ∈ ℝ | m < -9 }
• Interval notation: (-∞, -9)
Step 3. Graph the solution set
To graph m < -9 on a number line:
1. Draw a number line.
2. Mark -9 with an open circle (since -9 is not included).
3. Shade the portion to the left of -9, which represents all the values less than -9.
This represents the solution to the inequality.
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Simplify this solution Extra Insights
First, let's solve the inequality \(-3m - 4 > 23\). Begin by isolating \(m\) on one side of the inequality: 1. Add 4 to both sides: \[-3m > 23 + 4\] \[-3m > 27\] 2. Now, divide both sides by -3. Remember, when dividing by a negative number, we flip the inequality sign: \[m < -9\] Now, let's graph the solution set. On a number line, you would draw an open circle at -9 (indicating that -9 is not included in the solution) and shade everything to the left of -9. In set-builder notation, the solution set can be written as: \[\{ m \ | \ m < -9 \}\] In interval notation, the solution set is denoted as: \((-∞, -9)\)
