Chapter 1: Q. 15 (page 107)

Determine whether each implication that follows is true or
false. Use graphs to justify any implications that are true and
counterexamples for any implications that are false.
If $0 < | x - 0 | < 0.75,$ then $|x^{2} - 0| < 0.5 .$

Short Answer

Expert verified

The implication is false.

Step by step solution

Step 1. Given information.

The given statement is the following.

If $0 < | x - 0 | < 0.75$ then $|x^{2} - 0| < 0.5 .$

Step 2. Justification.

Substitute $x = - 0.72$ in first inequality.

$0 < | x - 0 | < 0.75 0 \overset{?}{<} | - 0.72 - 0 | \overset{?}{<} 0.75 0 < 0.72 < 0.75$

Substitute $x = - 0.72$ in second inequality.

role="math" $|x^{2} - 0| < 0.5 |{(- 0.72)}^{2} - 0| \overset{?}{<} 0.5 0.5184 ≮ 0.5$

The first inequality is satisfied at $x = - 0.72$ but the second inequality is not satisfied.

So the implication is false.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Determine whether each implication that follows is true or
false. Use graphs to justify any implications that are true and
counterexamples for any implications that are false.
If $0 < | x - 0 | < 0.75,$ then $|x^{2} - 0| < 0.5 .$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Justification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Calculus

Geometry

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Determine whether each implication that follows is true orfalse. Use graphs to justify any implications that are true andcounterexamples for any implications that are false.If 0<|x-0|<0.75,then x2-0<0.5.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Justification.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Calculus

Geometry

Theoretical and Mathematical Physics

Statistics

Study anywhere. Anytime. Across all devices.

Determine whether each implication that follows is true or
false. Use graphs to justify any implications that are true and
counterexamples for any implications that are false.
If $0 < | x - 0 | < 0.75,$ then $|x^{2} - 0| < 0.5 .$