Problem 61 Solve each inequality in Exercis... [FREE SOLUTION]

Chapter 1: Problem 61

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|2 x-6|<8$$

Short Answer

Expert verified

The solution to the inequality $|2x-6|<8$ is $-1<x<7$, which is represented as $(-1,7)$ in interval notation.

Step by step solution

Convert the absolute inequality to linear inequalities

Recall that $|a|<b$ is equivalent to $-b<a<b$. This means we can rewrite $|2x-6|<8$ as $-8<2x-6<8$.

Solve these inequalities

To solve $-8<2x-6<8$, first isolate $2x$ in the middle by adding 6 to all parts of your inequality which gives $-2<2x<14$. Then, divide all sides of the inequality by 2 to isolate $x$ which yields $-1<x<7$.

Plot the solution set on the number line

On a number line, plot the solutions $x=-1$ and $x=7$ as open points (because the inequality doesn't include the possibility of them being equal). Then, draw a line between these two points to show all the numbers that are included in the solution set.

Convert the solution into interval notation

Interval notation expresses the solution set in terms of intervals. Our solution set, expressed in interval notation is then $(-1,7)$.

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 Absolute Value Inequalities

Absolute value inequalities are a special type of inequality that involve the absolute value sign. The absolute value of a number is its distance from zero on the number line. Distances are always non-negative, so absolute values are as well.

When dealing with inequalities like $|a| < b$, it means the value inside the absolute value sign is less than a certain positive quantity $b$. This breaks down to

$-b < a < b$ This approach transforms the absolute inequality into two simpler linear inequalities. By understanding this concept, we've opened the path to solving the inequalities more directly.

Grappling with absolute value can often seem complex, but remembering its core purpose鈥攖o measure distance from zero鈥攃an help clarify its role in inequalities.

Solving Linear Inequalities

Once we convert an absolute value inequality into a pair of linear inequalities, we treat these like any other linear inequality. The goal is to isolate the variable, typically represented by $x$, on one side of the inequality sign.

Let's consider

$-8 < 2x - 6 < 8$ To solve, we first add 6 to each part, obtaining $-2 < 2x < 14$. Then, we divide all parts by 2, giving $-1 < x < 7$.
- Addition or subtraction can be used to bring constant terms to one side.
- Multiplication or division helps isolate the variable.
  
  Remember, when multiplying or dividing by negative numbers, the inequality signs reverse. However, in our example, division was by positive 2, so the inequality signs stayed pointed the same way. This procedure highlights how solving linear inequalities closely resembles solving basic algebraic equations, with a few added precautions.

Using Interval Notation

Interval notation offers a concise way of expressing a range of values, particularly useful in describing solution sets of inequalities.

For the inequality $-1 < x < 7$, the solution set in interval notation is

$(-1, 7)$ This indicates all values $x$ that lie between $-1$ and $7$ but do not include $-1$ or $7$ themselves, hence the use of open brackets.
- $[$ or $]$ brackets, include endpoints.
- $($ or $)$ parentheses, do not include endpoints.
  
  Understanding interval notation helps in quickly communicating solutions, especially when dealing with large ranges or multiple intervals. It's a useful shorthand, efficient for both writing and interpreting. Though it may look unfamiliar at first, practice and usage make it an intuitive tool in describing sets.

Visualizing with a Number Line

Graphing inequalities on a number line gives a visual representation of the solution set, making it easier to comprehend and verify the result. In our inequality solution $-1 < x < 7$, here's how it's visualized:

Mark -1 and 7 with open circles since they are not included.
Shade the region between these points to show all possible $x$-values in the solution set. The number line helps in visualizing which numbers satisfy the inequality. It's an effective method for conceptualizing the problem, especially when dealing with more complex inequalities.

Using number lines enables students to visually gauge the scope of the solutions, reinforcing understanding of where those solutions lie relative to each other and relative to critical points on the line.

91影视

Solve each inequality in Exercises 57-84 by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the solution set using interval notation. $$|2 x-6|<8$$

Short Answer

Step by step solution

Convert the absolute inequality to linear inequalities

Solve these inequalities

Plot the solution set on the number line

Convert the solution into interval notation

Key Concepts

Understanding Absolute Value Inequalities

Solving Linear Inequalities

Using Interval Notation

Visualizing with a Number Line

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Statistics

Pure Maths

Geometry

Decision Maths

Study anywhere. Anytime. Across all devices.