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Chapter 8: Problem 52

Graph the solution set, and write it using interval notation. $$ -15<3 x+6<-12 $$

Short Answer

Expert verified
(-7, -6)

Step by step solution

01

- Isolate the variable

The first step is to isolate the variable, which is \(x\) in this case. Start by solving the compound inequality \(-15 < 3x + 6 < -12\). To do this, subtract 6 from all parts of the inequality:-15 - 6 < 3x + 6 - 6 < -12 - 6This simplifies to:-21 < 3x < -18
02

- Further isolate \(x\)

Next, we divide each part of the inequality by 3 to solve for \(x\):\frac{-21}{3} < \frac{3x}{3} < \frac{-18}{3} This simplifies to:-7 < x < -6
03

- Write the solution in interval notation

The inequality \(-7 < x < -6\) can be written in interval notation as follows: (-7, -6)
04

- Graph the solution

To graph the solution set, draw a number line and shade the region between -7 and -6 (not including -7 and -6, since the inequalities are strict). Place open circles at -7 and -6 to show that these points are not included in the solution set.

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Key Concepts

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

Interval Notation
Understanding interval notation helps you express the range of solutions in a compact form. In our compound inequality \[ -7 < x < -6 \], we see that the solution is the set of all numbers between -7 and -6, but not including -7 and -6 themselves. This is because the inequality signs are strict (<), meaning these endpoints are not part of the solution.

To write this in interval notation, use parentheses:
  • A parenthesis \( ( \) or \( ) \) means that the endpoint is not included.
  • A bracket \( [ \) or \( ] \) means that the endpoint is included.
For this exercise, since -7 and -6 are not included, we write the solution as \( (-7, -6) \). This notation is both concise and precise, providing a clear way to express intervals.
Number Line Graphing
Graphing the solution set on a number line is a visual method to understand the range of values that satisfy the inequality. For \[ -7 < x < -6 \], you'll follow these steps:
  • Draw a horizontal line to represent the number line.
  • Mark the points -7 and -6 on the number line.
  • Place open circles at -7 and -6 to show that these points are not included.
  • Shade the region between -7 and -6.
By shading the area between -7 and -6 and using open circles, you're indicating that any value in this region is a valid solution, while -7 and -6 themselves are not.
Isolating Variables
Isolating the variable is a fundamental step in solving inequalities, ensuring that you solve for the variable correctly. Here, we start with the inequality \-15 < 3x + 6 < -12\. The goal is to isolate \( x \). Follow these steps:
  • First, subtract 6 from all parts of the inequality: \( -15-6 < 3x + 6 - 6 < -12 - 6 \). This simplifies to \( -21 < 3x < -18 \).
  • Next, divide each part by 3 to solve for x: \[ \frac{-21}{3} < \frac{3x}{3} < \frac{-18}{3} \]. This simplifies to \[-7 < x < -6 \].
Each of these operations keeps the inequality balanced, and it's important to perform them on all parts simultaneously.

The isolation process is not just about getting the variable alone but ensuring every step respects the inequality rules to maintain solution integrity.

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Most popular questions from this chapter

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