Problem 70 Use interval notation to express... [FREE SOLUTION]

Chapter 1: Problem 70

Use interval notation to express the solution set of each inequality. $$|2 x-5| \geq 1$$

Short Answer

Expert verified

The solution to the given absolute value inequality $|2x-5| \geq 1$ in interval notation is $(-\infty, 2] \cup [3, +\infty)$.

Step by step solution

Handle the Case Where $2x-5$ is Greater Than or Equal to 1

Firstly, we handle the case where $2x-5 \geq 1$. To isolate x, we add 5 to both sides of the equation to get $2x \geq 6$. Dividing by 2, the inequality becomes $x \geq 3$. The solution to this part of the inequality in interval notation is $[3, +\infty)$.

Handle the Case Where $2x-5$ is Less Than or Equal to -1

Secondly, the case $2x-5 \leq -1$ needs to be handled. We add 5 to both sides and get $2x \leq 4$. Dividing by 2, we simplify the inequality to $x \leq 2$. In interval notation, the solution to this part of the inequality is $(-\infty, 2]$.

Combine the Solutions

We combine the solutions from Steps 1 and 2. The common part of these intervals is the empty set, thus we write the full solution as the union of the two solution sets, which is $(-\infty, 2] \cup [3, +\infty)$.

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.

Absolute Value Inequality

An absolute value inequality like $|2x-5| \geq 1$ involves finding all values of $x$ that keep the expression within the absolute value above or equal to a certain number. Absolute value signifies the distance from zero on a number line, regardless of direction. In this inequality, there are two scenarios to consider:

The expression inside, $2x-5$, can be greater than or equal to 1.
Or it can be less than or equal to -1.

To solve, we separate these cases and solve each like a regular inequality:

For $2x-5 \geq 1$, we find $x \geq 3$.
For $2x-5 \leq -1$, we find $x \leq 2$.

These two intervals help us define where $2x-5$ meets each condition.

Interval Notation

Interval notation is a way to express sets of numbers along a number line. It uses brackets and parentheses to show where numbers are included or excluded.

Square brackets $[\, ]$ mean the number is included in the interval.
Parentheses $(\, )$ indicate the number is not included.

In the steps earlier, we found two solutions:

$x \geq 3$ is written as $[3, +\infty)$, meaning all numbers from 3 onward. The infinity symbol $+\infty$ always uses a parenthesis because it isn鈥檛 a finite number.
$x \leq 2$ is written as $(-,2]$, which includes all numbers up to 2. The parenthesis before $-\infty$ indicates that there's no starting point, and the square bracket after 2 includes it in the set.

This notation lets us quickly read and understand the range of numbers.

Union of Intervals

The union of intervals combines different sets of solutions into one. For example, the solutions we found

$(-, 2]$
$[3, +\infty)$

are combined using the symbol $\cup$, which represents a union. Writing these as $(-, 2] \cup [3, +\infty)$ means any number that fits in either interval is part of the solution. The union is used here because there鈥檚 no overlap between the two intervals, creating an effective way to keep both valid ranges as part of the solution. This visual combination is powerful because it clearly shows the full solution set without getting bogged down in the specifics of each inequality.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Use interval notation to express the solution set of each inequality. $$|2 x-5| \geq 1$$

Short Answer

Step by step solution

Handle the Case Where \(2x-5\) is Greater Than or Equal to 1

Handle the Case Where \(2x-5\) is Less Than or Equal to -1

Combine the Solutions

Key Concepts

Absolute Value Inequality

Interval Notation

Union of Intervals

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Calculus

Pure Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.